geometry problems from Brazilian Training Lists for Cono Sur, IberoAmerican and IMO with aops links in the names
collected inside aops: here
Cono Sur Training Lists
1999 - 2020
no problems from 2019
Let $ABCD$ be a convex quadrilateral, $M$ the midpoint of $BC$ and $N$ the midpoint of $AD$. Prove that
$$[ABMN] = [NMCD] \Leftrightarrow BC \parallel AD$$where $[ \,]$ denotes area.
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle DEA = 90^o$, $AB = CD = EA = 1$ and $BC + DE = 1$. Calculate the area of the pentagon.
Given two circles each external to the other one, we draw a common inner and outer tangent. The resulting tangency points define a chord in each circle. Prove that the point of intersection of the lines of these two chords is collinear with the centers of the circles.
Let $ABC$ be a triangle right in $A$, $I$ the incenter of $ABC$ and $O$ the midpoint of hypotenuse $BC$. Prove that $OI < IA$.
$A,B,C$ are three points on the circumference of a circle of radius $ r$, with A$B = BC$. $D$ is a point interior of the circle such that the triangle $BCD$ is equilateral. The line through $A$ and $D$ intersects the circle again at $E$. Prove that $DE = r$.
Let $ABC$ be a triangle right in $A$ and $P$ a point within $ABC$, such that $AB = PB$. If $H$ is the foot of the altitude on $BC$ and $M$ is the midpoint of $BC$, prove that $PM$ is bisector of $\angle HPC$ if and only if $\angle ABC = 60^o$
The angle between the side $AC$ of triangle $ABC$ and the median relative to vertex $A$ is equal to $30^o$. This is also the measure of the angle between side $BC$ and the median relative to vertex $B$. Prove that triangle $ABC$ is equilateral.
Show that it is not possible to cover a convex polygon with $n$ sides using only non convex quadrilaterals.
Let $A,B,C$ be three distinct points on the plane such that $AB = AC$. Find the locus of the points $P$ such that $\angle APB = \angle APC$.
The lengths of the sides of a right triangle are integers. The length of the hypotenuse is not divisible by $5$. Prove that the area of the triangle is a multiple of $10$.
Convex $ABCD$ is inscribed in a semicircle of diameter $AB$. Let $S$ be the intersection point of $AC$ and $BD$, and $T$ a point on A$B$ such that $TS$ is perpendicular to $AB$. Prove that $ST$ is the bisector of the angle $\angle CTS$.
Let $ABC$ be a triangle for which there is an interior point $O$ such that $\angle BAO =\angle CBO =\angle ACO = 30^o$. Prove that the triangle $ABC$ is equilateral.
Is it possible to cut a circle of paper into pieces and then rearrange the pieces to get a square of the same area?
Note: Only a finite number of cuts are allowed, and each cut is either a straight line segment or an arc of a circle.
The bisector of angle $A$ of $ABC$ intersects side $BC$ at point $U$. Prove that the perpendicular bisector of segment $AU$, the perpendicular on $BC$ from $U$ and the diameter of the circumscribed circle through $A$ are concurrent.
An quadrilateral $ABCD$ is special if it is inscribed and has perpendicular diagonals. Let $ABCD$ be a special quadrilateral , $T$ is the point intersection of the diagonals, $O$ is the center of the circumscribed circumference and $P$ is the midpoint of $BC$. Prove that:
(a) the line perpendicular on the side $AD$ from $T$ contains the point $P$.
(b) $OP =\frac12 AD$.
In the parallelogram $ABCD$ , let $O$ be the point of intersection of the diagonals $AC$ and $BD$. The angles $\angle CAB$ and $\angle DBC$ measures twice the angle $\angle DBA$ , and the angle $\angle ACB$ is $r$ times the angle $\angle AOB$ . Determine $r$.
Let $P$ be a point inside an angle that does not belong to its bisector. Two segments are considered through $Q$, $AB$ and $CD$, with $A$ and $C$ on one side of the angle, $B$ and $D$ on the other side of the angle, and such that $P$ is the midpoint of $AB$ and $CD$ is perpendicular to the angle bisector. Demonstrate that $AB > CD$.
Show how to construct with ruler and compass the triangle $ABC$ knowing the length$ a$ of the side $BC$, the measure of the angle $\angle BAC$ and the sum $d$ of the lengths of the other two sides of $ABC$.
The point $D$ on the side $BC$ of the triangle with acute angle $ABC$ is such that $AB = AD$. Let $E$ be the point on the altitude of $ABC$ drawn from $C$ such that the circle $a_1$, with center $E$ tangent to the line $AD$ at point $D$. Let $a_2$ be a circle passing through $C$ and tangent to line $AB$ at $B$. Prove that $A$ is on the line joining the intersection points of $a_1$ and $a_2$.
For a point $P$ inside a triangle of area $S$ we draw three lines, none of them passing through any of the vertices of the triangle, in such a way that each side of the triangle exactly intersects two lines. In this way the triangle is partitioned into six smaller polygons, three of which are triangles. Let $S_1, S_2, S_3$ be the areas of such smaller triangles, prove that$$\frac{1}{S_1}+\frac{1}{S_2}+\frac{1}{S_3} \ge \frac{18}{S}$$
The point $O$ is the center of the circle inscribed in a isosceles trapezoid $ABCD$, of bases $AB > CD$. Let $M$ be the midpoint of $AB$, $E$ be the point where the inscribed circle touches the side $CD$ and $F$ the intersection point of the line $OM$ with the base $CD$. Prove that $DE = CF \Leftrightarrow AB = 2CD$.
Determine the locus of the orthocenters of the inscribed triangles in a given circle.
In a triangle $ABC$ let $O$ be the circumcenter and $P, H$ respectively the feet of the angle bisector and altitude relative to BC. Construct $\vartriangle ABC$ with ruler and compass, knowning the positions of $O,P$ and $H$.
From the vertices of an $ABCD$ quadrilateral, we draw perpendicular to the diagonals $AC$ and $BD$. Prove that the feet of these perpendiculars are the vertices of a quadrilateral similar to the original quadrilateral.
Note: Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are similar with corresponding vertices $A \leftrightarrow A_1$, $B \leftrightarrow B_1$,$C \leftrightarrow C_1$ and $D \leftrightarrow D_1$ if the following two conditions are met:
$$\bullet \frac{AB}{A_1B_1}=\frac{BC}{B_1C_1}=\frac{CD}{C_1D_1}=\frac{DA}{D_1A_1}$$$$\bullet \angle A = \angle A_1, \angle B = \angle B_1, \angle C = \angle C_1, \angle D = \angle D_1$$
$AC'BA'CB'$ is a convex hexagon such that $AB'= AC$',$ BC'= BA'$, $CA' = CB'$ and $\angle A + \angle B +\angle C = \angle A' + \angle B' + \angle C'$. Prove that the area of the triangle $ABC$ is half the area of the hexagon.
PS. Hexagon had a typo as $AC'BACB'$ . I hope it is correct now.
Let $ABCD$ a rectangle and $E,F$ points in the segments $BC$ and $DC$, respectively, such that $\angle DAF=\angle FAE$. Show that if $DF+BE=AE$, then $ABCD$ is a square.
Let $ABCDEF$ be a convex hexagon and $A_1$,$B_1$, $C_1$, $D_1$, $E_1$, $F_1$ the midpoints of the sides are $AB$, $BC$, $CD$, $DE$, $EF$, $FA$, respectively. Find the area of the hexagon depending on the areas of the triangles $ ABC_1$, $BCD_1$, $CDE_1$, $DEF_1$, $EFA_1$ and $FAB_1$.
A semicircle with center $O$ and diameter $AB$ is given. A straight line intersects $AB$ at $M$ and the semicircle at $C$ and $D$, such that $MB < MA$ and $MD < MC$. The circumcircles of the triangles $AOC$ and $DOB$ intersect again at $K$. Show that the lines $MK$ and $KO$ are perpendicular.
Let $D$ be the midpoint of the base $AB$ of the acute isosceles triangle $ABC$. Pick a point $E$ on $AB$ and let $O$ be the circumcenter of the triangle $ACE$. Prove that the line that passes through $D$ and is perpendicular on $DO$, the line that passes through $E$ and is perpendicular on $BC$ and the line that passes through $B$ and is parallel to $AC$ are concurrent.
In the convex quadrilateral $ABCD$, $AB = BD$ and the angles $\angle BAC$ and $\angle ADC$ are $30^o$ and $150^o$, respectively. Prove that the angles $\angle BCA$ and $\angle ACD$ are equal.
Let $O$ be the circumcenter of an isosceles triangle $ABC$ with $AB = BC$. $M$ is any point of $BO$, $M'$ is symmetric of $M$ wrt the midpoint of $AB$, $K$ is the intersection of $M'O$ with $AB$ and $L$ is the point on the side $BC$ such that $\angle CLO = \angle BLM$ . Show that points $O, K, B$ and $L$ lie on the same circle.
Let $ABC$ be a triangle with $AC < BC$. Let $F$ be the foot of the altitude relative to the side $AB$, $H$ be the orthocenter of $ABC$ and $O$ be the circumcenter of $ABC$. Let also $P$ be the intersection of $AC$ with the perpendicular on $OF$ passing through $F$. Prove that $\angle PHF=\angle BAC$ .
Given a circle $\Gamma$, a line $d$ does not intersect. $M$ and $N$ are variable points on the line $d$ such that the circle with diameter $MN$ is externally tangent to $\Gamma$. Prove that there is a point $P$ on the plane such that, to any such segment $MN$, the angle $\angle MPN$ is fixed.
Let $M$ and $N$ be points on the side $BC$ of a triangle $ABC$ such that $BM = CN$, with $M$ between $ B$ and $N$. $P$ and $Q$ lie, respectively, on $AN$ and $AM$ such that $\angle PMC= \angle MAB$ and $\angle QNB = \angle NAC$ . Prove that $\angle QBC=\angle PCB$.
Let $ABCD$ be a convex quadrilateral such that $\angle ABC =\angle ADC = 135^o$ and $AC^2 \cdot BD^2 = 2 \cdot AB \cdot BC \cdot CD \cdot DA$. Prove that the diagonals of the quadrilateral are perpendicular .
Let $O$ be an interior point of the triangle cut-angle $ABC$. the circles with centers at the midpoints of the sides and passing through $O$ intersect again at points $K, L$, and $M$. Show that $O$ is the incenter of $KLM$ if and only if $O$ is the circumcenter of $ABC$.
Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $A'$ be the symmetric of $A$ wrt$ BC$, $D$ the point on $AC$ such that $AB = AD$ and $H$ the orthocenter of $ABC$. If l is the external bisector of $\angle BAC$, $M = A'D \cap l$ and $N = CH \cap l$, prove that that $AM = AN$
An acute triangle $ABC$ has circumradius $R$, inradius r and its largest Angle is $\angle BAC$. Let $M$ be the midpoint of $BC$ and $X$ the intersection of two lines tangent to the circumcircle of $ABC$ passing through $B$ and $C$. Prove that $\frac{r}{R}\ge \frac{AM}{AX}.$
Consider the circle $\Gamma$ with center $O$ and radius $R$. Let A be a point outside the $\Gamma$ . Draw a line $\ell$ by $A$, other than $AO$, cutting $\Gamma$ at $B$ and $F$, with $B$ between $A$ and$ F$. Let $\ell'$ be the symmetric line of $\ell$ with respect to $AO$. This line cuts $\Gamma$ at $C$ and $D$, with $C$ between $A$ and $D$. Show that the intersection of lines$ BD$ and $CF$ is independent of the choice of line $\ell$.
Let $\Gamma$ be a circle and $\ell$ a line that does not intersect it. be AB the diameter of $\Gamma$ perpendicular on $\ell$ , with $B$ closest to $\ell$ . A point $C$ other than $A$ ,$B$ is chosen over $\Gamma$. line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ in $E$, with $B$ and $E$ on the same side of $AC$. Let $F$ be the intersection of $BE$ with $\ell$ and $G$ the second intersection of $AF$ with $\Gamma$. Let $H$ also be the second intersection of $CF$ and $\Gamma$ . Show that $AB$ is perpendicular to $GH$.
Let $\omega$ be the incircle of the triangle $ABC$. Let $M$ be the midpoint of $BC$ and $D$ the intersection of $\omega$ with $BC$. Let $\omega_1$ be the circle of center M passing through $D.$ The circles $\omega_2$ and $\omega_3$ are constructed analogously. show that if $\omega_1$ is tangent to the circle $\Gamma$ of $ABC$, then $\omega_2$ or $\omega_3$ is also tangent to $\Gamma$ .
Let $H$ be the orthocenter of a triangle with an acute angle $ABC$ and $B', C'$ are projections of $B$,$C$ on $AC$ and $AB$, respectively. a straight line passing through $H$ intersects the segments $BC'$ and $CB'$ in $M$ and $N$, respectively. Lines perpendicular on $\ell$ passing through $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$, respectively. Find the locus of the midpoints of $PQ$ when line $\ell$ varies.
All diagonals of a regular $25$-sided polygon are drawn. Prove that there are no $9$ diagonals passing through a point inside the polygon.
Let $M$ be the intersection point of the diagonals of an cyclic quadrilateral $ABCD$, where $\angle AMB$ is acute. The isosceles triangle $BCK$ is constructed outside the quadrilateral, with the base being $BC$, such that $\angle KBC + \angle AMB = 90^o$. Prove that $KM$ is perpendicular to $AD$.
Let $ABC$ be a triangle with circumradius $R = 1$. Let $ r $ be the inradius of the triangle $ABC$ and $p$ be the inradius of the orthic triangle $A_1B_1C_1$. Show that $p \le 1- \frac{1}{3}(1+r)^2$,
Let $A_1$, $B_1$ and $C_1$ be the intersections of the internal bisectors of a triangle $ABC$ with sides $BC$, $CA$ and $AB$, respectively, and let $A_2$, $B_2$ and $C_2$ be the midpoints of segments $B_1C_1$, $C_1A_1$ and $A_1B_1$, respectively. Prove that $AA_2$, $BB_2$ and $CC_2$ are concurrent .
Let $ABC$ and $A'BC$ be two equilateral triangles ($A\ne A'$) and $D$ a variable point on the side $AC$ . Let $A'D$ intersects side $AB$ at $E$ and $BD$ intersects $CE$ at $P$. Show that $P$ lies on the circle circumscribed around the triangle $ABC$.
Segment $AB$ intersects two circles of the same radius, and is parallel to the line is connects the centers of these circles and all the intersection points of that segment with the circles is between $A$ and $B$. From $A$, we draw tangents to the circle closest to $A$, and from $B$, we draw tangents to the circle closest to $B$. It is verified that the quadrilateral formed by the four tangents contains both the circles. Prove that this quadrilateral is tangential.
Prove that if all the vertices of a polygon in the Euclidean plane are distant in the maximum $1$ of each other, then the area of this polygon is less than $\frac{\sqrt3}{2}.$
Let $ABCD$ be a convex quadrilateral. Consider the points $E =\overline{AD} \cap \overline{BC}$ and $I = \overline{AC} \cap \overline{BD}$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB\parallel CD$ and $IC^2 = IA \cdot AC$.
Let $ABCD$ be a convex quadrilateral. Consider the points $P = \overline{AB} \cap \overline{CD}$ and $Q = \overline{AD} \cap \overline{BC}$. Let $O$ be an interior point of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that $\angle AOB +\angle COD = 180^o$.
A point $P$ is inside a triangle $ABC$. The side $AC$ meets the line $BP$ at $Q$ and $AB$ meets $CP$ at $R$. Suppose that $AR = RB = CP$ and $CQ = PQ$. Find $\angle BRC$.
An acute triangle has all its sides of different lengths . Show that the straight line passing through its circumcenter and its incenter intersects the longest and shortest sides of the triangle.
The altitudes $AD$ and $BE$ of triangle $ABC$ meet at orthocenter$ H$. The midpoints of $AB$ and $CH$ are $X$ and $Y$ , respectively. Prove that $XY$ is perpendicular to $DE$
Consider the triangle $ABC$ whose lengths of the sides are $a, b, c$, where $a$ is the longest side. Prove that $ABC$ is right if, and only if,$$(\sqrt{a+b}+\sqrt{a-b})(\sqrt{a+c}+\sqrt{a-c})=(a+b+c)\sqrt2$$
A circle with center $I$ is inside another circle. $AB$ is a variable chord of the large circle that is tangent to the small circle. Determine the locus of the circumcenter of the triangle $ABI$.
Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$ and such that $AC \ne BC$. The lines $CH$ and $CI$ cut the circumcircle of $ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^o$ if, and only if, $\angle IDL = 90^o $.
Within a unit square, three points are chosen. Show that the distance between two of them is not greater than $\sqrt6-\sqrt2$.
Let $ABCD$ be a parallelogram. the bisector of $\angle BAD$ cuts $BC$ at $M$ and cuts the extension of $CD$ at $N$. The circumcenter of the triangle $MCN$ is $O$. Prove that $B, O, C, D$ they are concentric.
Let $D$ be a point on the side $BC$ of the acute triangle $ABC$ ($D \ne B$ and $D \ne C$), $O_1$ the circumcenter of the triangle $ABD$, $O_2$ is the circumcenter of triangle $ACD$ and $O$ is the circumcenter of triangle $AO_1O_2$. Determine the locus of point $O$.
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$ and $\angle PAC = \angle PCB$. the point $Q$ is on the straight line $BC$ satisfying $QA = QP$. Prove that $\angle AQP =2\angle OQB$.
A game is played by two players on an infinite plane containing $51$ pieces, $50$ sheep and $ 1$ wolf. The first player starts the game by moving the wolf. Then the second player moves some sheep, and so on. On each move, a piece can move up to a distance of $ 1$ meter. Is it true that, regardless of the initial position of the pieces, the wolf always manages to capture at least one sheep?
Let $ABC$ be a triangle with circumcircle $\omega$ such that $\angle BAC = 60^o$. Given a point $ B$ on the arc $BC$ of $\omega$, let $O_1$, $O_2$ the circumcenters of the triangles $ABD$ and $ACD$, respectively, $M$ the intersection of $BO_1$ and $CO_2$ and $N$ the circumcircle of $DO_1O_2$. Show that, when varying $D$, $MN$ passes through a fixed point .
Let $\omega$ be the incircle of a triangle ABC. The median $AM$ of $ABC$ intersects $\omega$ at $K$ and $L$. The lines parallel to $BC$ passing through $K$ and $L$ intersect $\omega$ again at $X$ and $Y$. Let $P, Q$ be the intersections of $BC$ with the lines $AX$ and $AY$ . Show that $BP = CQ$.
Let $ABC$ be a triangle with circumcircle $\omega_1$, $O$ the circumcenter of $ABC$ and $\omega_2$ the $A$-excircle . If $M, N, L$ are the touchpoints of $\omega_2$ with the lines $BC, AC, AB$ repsectively and the radii of $\omega_1$ and $\omega_2$ are equal, show that $O$ is the orthocenter of the triangle $MNL$ .
Let $ABCDE$ be a pentagon with $AE = ED$, $AB + CD = BC$ and $\angle BAE + \angle CDE = 180^o$. prove that $\angle AED = 2\angle BEC$.
Let $ABC$ be a triangle and $D$ the point on the extension of side $BC$ beyond $ B$ such that $BD = BA$, and let $M$ be the midpoint of $AC$. The angle bisector of $\angle ABC$ cuts $DM$ to $P$ . Prove that $\angle BAP = \angle ACB$.
In the forest where the Smurfs live, Gargamel planted $1280$ pine trees, each $1$ meter in diameter. The forest is a rectangular field of dimensions $1001\times 945$ meters. Grandpa Smurf would like to build seven tennis courts , each measuring $20\times 34$ meters. Grandpa Smurf will be able to do the construction without knocking down no trees, no matter how Gargamel has planted the trees?
Consider five segments in the plane so that any three of them form a triangle. Prove that at least one of these triangles is acute.
Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$ such that $\angle BAC = 60^o$ and $AB > AC$. Let $BE, CF$ be the the altitudes of $ABC$, and $M,N$ points on the segments $BH, HF$ respectively, such $BM = CN$. Determine the value of the expression $\frac{HM + NH}{OH}.$
Let $ABCD$ be a convex quadrilateral, with $AB$ not parallel to $CD$, and let $X$ be a point within $ABCD$ such that $\angle ADX = \angle BCX < 90^o$ and $\angle DAX = \angle CBX < 90^o$. If the perpendicular bisectors of segments $AB$ and $CD$ intersect in $Y$ , prove that $\angle AYB = 2\angle ADX$
Consider a point $P$ on the outside of a circle $\omega$. The two tangent lines to $\omega$ starting from $P$ intersect $\omega$ at points $A$ and $ B$. Let $M$ be the midpoint of $AP$ and $N$ the intersection of $BM$ with $\omega$, prove that $PN = 2MN$.
Let $ABC$ be an acute triangle such that $AB > AC$ and $\angle BAC = 60^o$. Let $O, H$ be the circumcenter, orthocenter of $ABC$, respectively. Knowing that $OH$ intersects $AB$ at $P$ and $AC$ at $Q$, prove that $PO = HQ$.
On the rhombus $ABCD$ with $\angle A = 60^o$, we take points $F,H$ and $G$ on sides $AD$, $DC$ and diagonal $AC$ so that $DFGH$ is a parallelogram. Prove that the triangle $FBH$ is equilateral .
Given a triangle $ABC$ such that $\angle BCA = 60^o$ with circumcenter $O$ and incenter $I$. Let $A_1$ and $B_1$ be the feet of the internal bisectors of angles $\angle CAB$ and $\angle ABC$, respectively. The line $A_1B_1$ intersects the circumcircle of $ABC$ at $A_2$ and $B_2$.
(a) Prove that $OI$ is parallel to $A_1B_1$.
(b) If $R$ is the midpoint of the arc $AB$ not containing $C$ and $P , Q$ are the midpoints of $A_1B_1$, $A_2B_2$, respectively, prove that $RP = RQ$.
Given a circle $\omega$ of center $O$ and a point $P$ outside the $\omega$. Let $\ell_1$, $\ell_2$ be lines through $P$ such that $\ell_1$ touches $\omega$ in $A_1$ and $\ell_2$ interserts $\omega$ in $B$ and $C$. If the tangent lines of $\omega$ passing through $B$ and $C$ intersect at $X$, prove that the segments $AX$ and $PO$ are perpendicular.
Let $ABC$ be an acute triangle with alttiudes $AD$ $BE$ and $CF$ . Let $M$ be the midpoint of the segment $BC$. The circle circumscribed around the triangle $AEF$ cuts the line $AM$ at $A$ and $X$. Line $AM$ cuts line $CF$ at $Y$ . be $Z$ the intersection point of lines $AD$ and $BX$. Show that the lines $YZ$ and $BC$ are parallel.
In a triangle $ ABC$, we take points $X, Y$ on sides $AB$, $BC$, respectively. If $AY$ and $CX$ intersect at $Z$ and $AY = YC$ and $AB = ZC$, prove that points $B, X, Z, Y$ are concyclic,
The incircle of the acute triangle $ABC$ is tangent to the sides $AB, BC, CA$ at points $P,Q,R$, respectively. Suppose the orthocenter $H$ of $ABC$ belongs to the segment $QR$ .
(a) Prove that $PH \perp QR$.
(b) Let $I, O$ be the incenter, circumcenter of $ABC$, respectively, and $N$ is the touch point of $AB$ with the $C$-excircle , prove that $I, O, N$ are collinear.
Construct a triangle $ABC$, knowing the positions of:
$\bullet$ $O$, the circumcenter of $ABC$
$\bullet$ $D$, the feet of altitude from vertex $A$
$\bullet$ $P$, the feet of the internal bisector of angle $\angle BAC$
Let $K$ and $L$ be points on sides $AB$ and $AC$, respectively, of the triangle $ABC$ such that $BK = CL$. Let P be the intersection point of the segments $BL$ and $CK$, $M$ a point inside the segment $AC$ such that $MP$ is parallel to the bisector of the angle $\angle BAC$. Prove that $CM = AB$.
Let $ABC$ be a triangle, and let $D, E$ points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are respectively , the touchpoints with $BC$ of the incircles of the triangles $ABD$ and $ACE$, prove that$$ \frac{1}{BM}+ \frac{1}{MD}= \frac{1}{NC} +\frac{1}{NE}.$$
Let $ABC$ be an acute triangle with circumcenter $O$. $T$ is the circumcenter of the triangle $AOC$ and $M$ is the midpoint of the side $AC$ . On sides $AB$ and $BC$, points $D$ and $E$ are taken, respectively, such that $\angle BDM = \angle BEM=\angle ABC$. Show that the lines $BT$ and $DE$ are perpendicular.
Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $ABC$, respectively, so that $\angle AXY =2\angle ACB$ and $\angle CYX = 2\angle BAC$. Prove that$$\frac{S(AXYC)}{S(ABC)} \le \frac{AX^2 + XY^2 + YC^2}{AC^2}$$where $S$ denotes the area of the corresponding figure.
All sides of a convex pentagon have the same length and each of the angles is less than $120^o$. Prove that all angles are not obtuse.
Is it possible for us to cover a cube with three paper triangles (without overlapping)?
A right triangle $ABC$, with hypotenuse $AB$, is inscribed in a circle $\Gamma$. Let $K$ be the midpoint of the arc $BC$ that does not contain $A$, $N$ be the midpoint of the side $AC$ and $M$ be the point of intersection of $KN$ with the circle $\Gamma$. Let $E$ be the intersection point of the tangents to the circle at A and C. Prove that $\angle EMK = 90^o.$
Show that we cannot draw two triangles of area $1$ within a circle of radius $1$ so that the triangles don't have a common point.
An integer coordinate point is a point $(x, y)$ with both coordinates integer . Find the smallest $n$ such that every set with $n$ integer coordinate points has three points that are vertices of a triangle with integer area . (The triangle can be degenerate, in other words, the three points can be on the same line forming a triangle of area zero).
The circles $G_1$ and $G_2$ intersect at two points, $A$ and $B$. A line through $B$ intersects $G_1$ at $C$ and $G_2$ at $D$. A tangent line to $G_1$ that passes through $C$ cuts the tangent line from $G_2$ that passes through $D$ in $E$. The symmetrical line of $AE$, wrt line $AC$, cuts $G_1$ at $F$ (besides $A$). . Prove that $BF$ is tangent to $G_2$.
Note: Consider only the case where $C$ does not belong to the interior of $G_2$ and $D$ does not belong to the interior of $G_1$.
Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ intersect at $A$ and $B$. Let $S_3$ be a circle passing through $O_1$, $O_2$ and $A$. The circle $S_3$ intersects $S_1$ at $D \ne A$, $S_2$ at $E$ and the line $AB$ in $C$. Prove that $CD = CB = CE$.
A disc of radius $ 1$ is covered by $7$ identical discs of radius $r$. Prove that $ r \ge \frac12$.
Let $ABCD$ be a cyclic quadrilateral. Prove that$$|AC - BD| \le |AB - CD|.$$When does equality occur?
Let $ABC$ be a non-equilateral triangle whose incircle touches $BC$, $CA$ and $AB$ at $A_1$,$B_1$ and $C_1$, respectively. Let $H_1$ be the orthocenter of the triangle $A_1B_1C_1$ . Prove that $H_1$ lies on the straight line passing through the incenter and circumcenter of the triangle $ABC$.
Points $A_1$,$B_1$ and $C_1$ lie on sides $BC$; $CA$ and $AB$ of the isosceles triangle $ABC$ ($AB = BC$). Knowing that $\angle BC_1A_1 = \angle CA_1B_1 = \angle BAC$ and $BB_1 \cap CC_1 = P$, prove that $AB_1PC_1$ is cyclic.
The quadrilateral $ABCD$ is inscribed in a circle $\omega$ with center $O$. The bisector of $\angle ABD$ meets $AD$ and $\omega$ at points $K$ and $M$, respectively. The bisector of $\angle CBD$ meets $CD$ and $\omega$ at points $L$ and $N$, respectively. Suppose $KL\parallel MN$. Prove that the circumcircle of the triangle $MON$ passes through midpoint of $BD$.
Show that for any convex polygon of area $1$ there is a parallelogram of area $2$ that contains it.
Let $ABC$ ($AC \ne BC$) be a triangle with the angle $\angle ACB$ acute and $M$ the midpoint of $AB$. Consider the point $P$ on the segment $CM$ so that the bisectors of the angles $\angle PAC$ and $\angle PBC$ intersect at the point $Q$ of $CM$. Find the angles $\angle APB$ and $\angle AQB$.
The circle inscribed with the triangle $ABC$ is tangent to sides $AB$, $BC$ and $CA$ at points $F, D$ and $E$, respectively. The segment $AD$ is bisected at $X$ by the inscribed circle, i.e., $AX = XD$. If $XB$ and $XC$ cut the circle at $Y$ and $Z$, respectively, prove that $EY = FZ$.
Let $ABC$ be an acute triangle and $D$ a point on the side $AB$. The circumcircle of the triangle $BCD$ intersects the side $AC$ at $E$. The circumcircle of the triangle $ADC$ intersects the side $BC$ at $F$. Let $O$ be the circumcenter of the triangle of the triangle $CEF$. Prove that the points $D$ and $O$ and the circumcenters of the triangles $ADE$, $ADC$, $DBF$ and $DBC$ are concyclic and the line $OD$ is perpendicular to $AB$
The convex set $F$ does not cover a semicircle of radius $R$. It is possible that two sets congruent to $F$ cover a circle of radius $R$? What if $F$ is not convex? ´
Let $ABCD$ be a cyclic quadrilateral and let $U$ be the intersection point of the sides $AB$ and $CD,$ and $V$ the intersection point of the sides $BC$ and $DA$ . The straight line starting from $V$ that is perpendicular on the bisector of the $\angle AUD,$ cuts the segments $UA$ and $UD$ at $X$ and $Y$ , respectively. Prove that $AX \cdot DY = BX \cdot CY$.
Consider a triangle $ABC$. Let $D$ be the foot of the altitude from the vertex $A$ and let $E ,F$ be points on the sides $AB$, $AC$ respectively such that $\angle ADE = \angle ADF$. Prove that the lines $AD$, $BF$, and $CE$ are concurrent.
Let $BE$ and $CF$ be the altitudes of the triangle $ABC$. Prove that $AB = AC$ if and only if $AB + BE = AC + CF$.
Let $ABC$ be a scalene triangle. Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ respectively, such that$$\frac{BA_1}{BC} =\frac{CB_1}{CA} =\frac{AC_1}{AB}.$$Prove that if the triangles $AB_1C_1$, $BC_1A_1$, $CA_1B_1$ have the same circumradius, then $A_1, B_1, C_1$ are the midpoints of triangle $ABC$
In an acute triangle $ABC$, $CF$ is alttiude and $BM$ is median. If $BM = CF$ and $\angle MBC = \angle FCA$, prove that $ABC$ is equilateral.
Consider two circles $\omega_1$ and $\omega_2$ that intersect at two points $A$ and $B$. Let $\ell $be a straight line passing through $B$ that intersects $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively. The tangent to $\omega_1$ passing through $C$ intersects the tangent to $\omega_2$ passing through $D$ at $E$. If the symmetric line of $AE$ wrt $AC$ intersects $\omega_1$ at $F$, $F \ne A$, show that $BF$ is tangent to $\omega_2$.
Given a triangle $ABC$, let $D$ and $E$ be interior points of sides $AB$ and $AC$, respectively, such that $B, D, E, C$ are concyclic. Let $F$ be the intersection of the lines $BE$ and $CD$. Circles circumscribed to the triangles $ADF$ and $BCD$ intersects at $G$ and $D$. Show that the line $GE$ intersects the segment $AF$ at its midpoint.
Squares $ABDE$ and $ACFG$ are constructed externally to an acute triangle $ABC$. If $H $is the orthocenter of $ABC$, show that the lines $AH$, $BF$ and $CD$ are concurrent.
Let $\vartriangle ABC$ be isosceles and right with hypotenuse $AB = \sqrt2$. Determine the positions of the points $X,Y,Z$ on the sides $BC$, $CA$, $AB$, respectively, such that the triangle $\vartriangle XYZ$ is right and isosceles with minimum area.
The quadrilateral $ABCD$ circumscribes a given circle $S$. Touchpoints of $S$ with sides $AB$, $BC$, $CD$ and $AD$ are points $E$, $F$, $G$ and $H$, respectively. The intersection of $AC$ and $BD$ is $I$. Prove that$$\frac{AI}{IC} = \frac{AH}{CD}.$$
Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$ and $AC \ne BC$. The lines $CH$ and $CI$ meet again the circumcircle of the triangle $\vartriangle ABC$ at the points $D$ and $L$, respectively. Prove that $\angle CIH = 90^o$ if, and only if, $\cos \angle A + \cos \angle B = 1$.
In the triangle $\vartriangle ABC$, satisfying $AB + BC = 3AC$, the incircle has center $I$ and is tangent to sides $AB$ and $BC$ at $D$ and $E$, respectively. Let $K$ and $L$ be the symmetric points of $D$ and $E$ wrt $I$. Prove that the quadrilateral $ACKL$ is cyclic.
Let $ABCD$ be a parallelogram. A variable line $\ell$ passing through $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$ , respectively. Let $K$ and $L$ be the centers of the excircles of the triangles $ABX$ and $ADY$ , which touch the sides $BX$ and $DY$ , respectively. Prove that the size of the angle $ \angle KCL$ does not depend on the choice of the line $\ell$.
The median $AM$ of the triangle $\vartriangle ABC$ intersects its incircle $\omega$ at $K$ and $L$. The lines through $K$ and $L$ parallel to $BC$ intersect $\omega$ again at $X$ and $Y$ . The lines $AX$ and $AY$ intersects $BC$ at $P$ and $Q$. Prove that $PB = CQ$.
Let $ABC$ be a triangle such that $\angle A = 90^o$ and $\angle B < \angle C$. Let $D$ be the intersection of the tangent by $A$ to the circumcircle of $ABC$ and the straight $BC$. Let $E$ be the symmetric of $A$ wrt $BC$ and $X$ be the foot of the perpendicular from $A$ on $BE$. Let $Y$ be the midpoint of $AX$ and $Z$ be the second intersection of $BY$ and the circumcircle of $ABC$. Prove that $BC$ is tangent to the circumcircle of $ADZ$.
Let $I$ be the incenter of a triangle $ABC$ with $AB \ne AC$. The lines $BI$ and $CI$ intersect sides $AC$ and AB at points $D$ and $E$, respectively. Find $\angle BAC$, knowing that $DI = EI$.
Let $P$ be a point inside the triangle $ABC$. the lines $BP$, $CP$ intersect $AC$, $AB$ at $Q$, $R$, respectively. Knowing that $AR = RB = CP$ and $CQ = PQ$, find the measure of the angle $\angle BRC$.
(a) Five identical paper triangles are given over a table. Each of them can move in any direction parallel to itself (i.e. without rotation). Prove or disprove: each triangle can be covered by the other four.
(b) Suppose the triangles in (a) are equilateral. Show that each triangle can be covered by the other four.
Let $ABC$ be a triangle and $ P$ be a point on the plane such that the triangles $PAB$, $PBC$ and $PCA$ have the same area and the same perimeter. Prove the statements below:
(a) If $ P$ lies inside $ABC$, then $ABC$ is equilateral. ´
(b) If $ P$ lies outside of $ABC$, then $ABC$ is right.
Let $ABC$ be a triangle, right at $ C$. The internal bisectors $AA_1$ and $BB_1$ intersect at incenter $I$. If $O$ is the circumcenter of the triangle $A_1B_1C$, prove that $OI$ and $AB$ are perpendicular.
Let $M$, $N$ and $ P$ be the intersection points of the incircle of the $\vartriangle ABC$ with the sides $AB$, $AC$ and $BC$, respectively. Prove that the intersection points of the altitudes of the $\vartriangle MNP$, the circumcenter of the $\vartriangle ABC$, and the incenter of the $\vartriangle ABC$ are collinear.
In the triangle $ABC$, let $J$ be the center of the excircle that is tangent to side $BC$ in $A_1$ and to the extensions of sides $AC$ and $AB$ in $B_1$ and $C_1$, respectively. Suppose lines $A_1B_1$ and AB are perpendicular and intersect at ̃ $D$. Let $E$ be the foot of the perpendicular of ́ $C_1$ n the line $DJ$. Determine the angles $\angle BEA_1$ and $\angle AEB_1$.
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be points in $ABCD$ such that $PQDA$ and $QPBC$ are cyclic quadrilaterals. Suppose there is a point $E$ on the segment $PQ$ such that $\angle PAE = \angle QDE$ and $\angle PBE = \angle QCE$. Show that the quadrilateral $ABCD$ is cyclic.
The triangles $\vartriangle ABC$ and $\vartriangle A'B'C'$ are the same with $AB = A'B'$ , $AC = A'C'$ and $BC$ = $B'C'$ but they have opposite orientations. Prove that the midpoints of ́ $AA'$ , $BB'$ and $CC'$ are collinear.
Let $ABC$ be an acute triangle and $M$ be the midpoint of ́ $BC$. There is only one interior point $N$ such that $\angle ABN = \angle BAM$ and $\angle ACN = \angle CAM$. Prove that $\angle BAN = \angle CAM$.
Let $ABCDE$ be a convex pentagon in the coordinate plane. Each of its vertices is an lattice point. The five diagonals of $ABCDE$ form a convex pentagon $A_1B_1C_1D_1E_1$ within ABCDE. Prove that this smaller pentagon contains a lattice point on its edge or interior.
A circle is inscribed in the isosceles trapezoid $ABCD$. Let $K$ and $L$ be the intersection points of the circle with $AC$ (with $K$ between $A$ and $L$). Find the value of$$\frac{AL \cdot KC}{AK \cdot LC}.$$
Let $a, b$, and $c$ be the lengths of the sides of a triangle and let $p$ the semiperimeter and $ r$ the inradius of the triangle. Prove that
$$\frac{1}{(p-a)^2}+\frac{1}{(p - b)^2}+\frac{1}{(p-c)^2}\ge \frac{1}{r^2}$$
The triangle $ABC$ has orthocenter $H$. The projections of $H$ on the internal and external bisectors of the angle $\angle BAC < 90^o$ are $ P$ and $Q$, respectively. Prove that $PQ$ passes through the midpoint ́of $BC$.
Let $ABCDE$ be a convex pentagon such that: ́ $\angle BAC = \angle CAD = \angle DAE$ and $\angle ABC =\angle ACD = \angle ADE$. Diagonals $BD$ and $CE$ intersect at $P$. Prove that $AP$ bisects the side $CD$.
Let $ABC$ be a triangle inscribed in a circle of radius $R$ and let $ P$ be a point inside it. Prove that$$\frac{PA}{BC^2} +\frac{PB}{CA^2}+\frac{PC}{AB^2} \ge \frac{1}{R}.$$
Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. On sides $AB$ and $AC$, consider the points $M$ and $N$, respectively, both different from the midpoints on the sides, such that $AM^2 + AN^2 = BM^2 + CN^2$ and $\angle MDN = \angle BAC$. Prove that $\angle BAC = 90^o$.
The acute triangle $ABC$ is given with $\angle BAC = 30^o$. The altitudes $BB_1$ and $CC_1$ are drawn. Let $B_2$ and $C_2$ be the midpoints of$ AC$ and $AB$, respectively. Prove that the segments $B_1C_2$ and $B_2C_1$ are perpendicular.
Given a scalene triangle $ABC$, be $A'$, $B'$, $C'$ the intersection points of the internal angle bisectors of $A$, $B$, $C$ with opposite sides, respectively. Let $A'''$ be the intersection of ̃ $BC$ and the perpendicular bisector of $AA'$. Define $B''$ and $C''$ analogously. Show that $A''$ , $B''$ and $C''$ are collinear.
Let $O$ be the circumcenter of an acute $\vartriangle ABC$ and $A_1$ a point on the smallest arc $BC$ of the circumcircle of $\vartriangle ABC$ . Let $A_2$ and $A_3$ be points on sides $AB$ and $AC$ respectively, such that $ \angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$. Demonstrate that line $A_2A_3$ passes through the orthocenter of the $\vartriangle ABC$.
Two circles intersect at $2$ points $A$ and $B$. A straight line that passes through point $A$ cuts the two circles at points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of arcs $BC$ and $BD$ (which do not contain point $A$) on their respective circles. Let $K$ be the midpoint of the segment ́ $CD$. Prove that $\angle MKN = 90^o$
Let $\vartriangle ABC$ be an acute triangle with $AB \ne AC$, let $V$ be the intersection of the internal bisector of the angle $\angle A$ with $BC$ and let $D$ be the foot of the altitude drawn from $A$. If $E$ and $F$ are the intersections of ̃circumcircle of the triangle $\vartriangle ADV$ with the lines $CA$ and $AB$, respectively, prove that the lines $AD$, $BE$ and $CF$ are concurrent.
Consider five points on the plane so that each triangle with vertices in these three points has an area less than or equal to $ 1$. Prove that the five points can be covered by a trapezoid of area at most $3$.
Let $ABCD$ be a rhombus and $P$ a point on the side $BC$ . The circle passing through $A$, $B$ and $P$ intersects the line $BD$ once more at point $Q$ and the circle passing through $C$, $P$ and $Q$ intersects $BD$ once more at point $R$. Prove that $A$, $R$, and $P$ are collinear points.
In the triangle $\vartriangle ABC$, $\angle ABC = \angle ACB = 40^o$ . Let $D$ be the point on $AC$ such that $BD$ and bisector of $\angle ABC$. Prove that $BD + DA = BC$.
Let $R$ be the radius of the circle circumscribed to the triangle $\vartriangle ABC$ of sides $a$, $ b$, and $c$. If $R=\frac{a\sqrt{bc}}{b + c}$ , find the angles of the triangle.
Let $k$ be the inscribed circle of a triangle ́$ABC$ not isosceles and let $I$ be the center of $k$. The circle $k$ touches sides $BC$, $CA$, and $AB$ at points $P, Q$, and $R$, respectively. Line $QR$ meets $BC$ at point $M$. Consider a circle $k'$ that contains the points $ B$ and $C$. Let $N$ be the intersection of $k$ and $k'$. The circumcircle of the triangle $MNP$ intersects the line $AP$ at point $L$, different from $P$. Prove that points $I, L$ and $M$ are collinear.
The circles $k$ and $k'$ with centers $O$ and $O'$ , respectively, are externally tangent at point $D$ and internally tangent to a circle $k''$ at points E and F, respectively. The line $t$ is common tangent of $k$ and $k'$ passing through $D$. Let $AB$ be the diameter of $k''$ perpendicular to $t$, such that $A, E$, and $O$ are on the same side wrt $t$. Prove that the lines $AO$, $BO'$, $EF$ and $t$ are concurrent.
Consider the point $O$ inside the triangle $ABC$ such that $\angle AOB = \angle BOC =\angle COA = 120^o$.
Prove that$$\frac{AO^2}{BC} +\frac{BO^2}{CA} +\frac{CO^2}{AB} \ge \frac{AO + BO + CO}{\sqrt3}$$
Given an acute triangle $ABC$, let $D$ be a point on side $BC$. Let $M_1$, $M_2$, $M_3$, $M_4$ and $M_5$ are the midpoints of the segments $AD$, $AB$, $AC$, $BD$ and $CD$, respectively. Let $O_1$, $O_2$, $O_3$ and $O_4$ be the circumcenters of the triangles $ABD$, $ACD$, $M_1M_2M_4$ and $M_1M_3M_5$, respectively. If $S$ and $T$ are the midpoints of segments $AO_1$ and $AO_2$, respectively, prove that $SO_3O_4T$ is an isosceles trapezoid.
The circles $k_0, k_1, k_2, k_3$ and $k_4$ are in a plane such that for $i = 1, 2, 3, 4$ a circle $k_i$ is tangent externally to $k_0$ at point $T_i$ and $k_i$ is tangent externally to $k_{i+1}$ at point $S_i$ ($k_5 = k_1$). Let $O$ be the center of $k_0$. The lines $T_1T_3$ and $T_2T_4$ intersect at the point $T$ and lines $S_1S_3$ and $S_2S_4$ intersect at point S. Prove that points $O, T$ and $S$ are collinear.
Calculate the minimum possible value of the perimeter of the triangle $ABC$, knowing that:
$\bullet$ $\angle A = 2 ( \angle B)$
$\bullet$ $\angle C > 90^o$
$\bullet$ sides $a, b, c$ of the triangle are integers.
Given a triangle $ABC$ and a point $ P$ inside it, let $D$ be the other intersection point of the line $BP$ with the circumscribed circle of the triangle $ABC$ and $Q$ the orthogonal projection of $P$ on the side $AC$ . Knowing that $\angle PAD = \angle APQ$ and $\angle PCD = \angle CPQ$ , prove that BQ is bisector of the angle $\angle ABC$.
Let H be the orthocenter of an acute triangle $ABC$. Show that the triangles $ABH$, $BCH$ and $CAH$ has the same perimeter if and only if the triangle $ABC$ is equilateral.
Let $Q$ be a point on the circle of diameter $AB$, where $Q$ is different from $A$ and $B$. Let $QH$ be the straight line perpendicular to $AB$ that passes through $Q$, where $H$ belongs to $AB$. the points of intersection of the circle of diameter $AB$ and the circle of center $Q$ and radius $QH$ are $C$ and $D$. Prove that $CD$ passes through the midpoint of $QH$.
Let $O$ be the circumcenter of the acute triangle ABC. Let $\Gamma$ be the circle that passes through points $A, B$ and $O$. Lines $CA$ and $CB$ intersect $\Gamma$ again at points $D$ and $E$. Prove that $CO$ perpendicular to $DE$.
Let $BC$ be the diameter of a semicircle and $A$ the midpoint of the arc. Let $M$ be a point on $AC$ and let $ P$ and $Q$ be the feet of the perpendiculars from $A$ and $C$ on the line $BM$, respectively. Prove that $BP = PQ + QC$.
The medians $AD, BE$ and $CF$ of the $ABC$ triangle intersect at the point $G$. Given that the quadrilaterals $AFGE$ and $BDGF$ are cyclic, prove that the triangle $ABC$ is equilateral.
Given the acute triangle $ABC$, let $M$ and $N$ be points on sides $AB$ and $AC$, respectively. The circles of diameter $BN$ and $CM$ intersect at points $P$ and $Q$. Prove that $P, Q$ and the orthocenter $H$ are collinear.
Consider an acute triangle $ABC$ with $AB \ne AC$. Let $D$ be the foot of the internal bisector starting from $A$ and let $E$ and $F$ be the feet of the altitudes drawn the vertices $ B$ and $C$, respectively. The circumcircles of the triangles $DBF$ and $DCE$ intersect again at point $M$, with $M \ne D$. Prove that $ME = MF$.
A rectangle $ABDE$ is drawn over side $AB$ of the acute triangle $ABC$ so that $C$ is over the side $DE$. Similarly, the rectangles $BCFG$ and $CAHI$ are defined, i.e. $ A$ is on the side $FG$ and $B$ is on the side $HI$ . The midpoints of $AB$, $BC$ and $CA$ are $J, K$ and $ L$, respectively. Prove that the sum of the angles $\angle GJH$, $\angle IKD$ and $ELF$ is $180^o$.
Let $ABC$ be an isosceles triangle such that $CA = CB$, let $O$ its circumcenter and $I$ its incenter. Consider a point $D$ on side $BC$ such that $DO$ is perpendicular to $BI$. Prove that $DI$ is parallel to $AC$.
Let $AH_a$ and $BH_b$ be the altitudes of the acute triangle $ABC$. The points$ P$ and $Q$ are the projections of $H_a$ on the sides $AB$ and $AC$. Prove that the line $PQ$ cuts the segment $H_aH_b$ at its midpoint.
Consider a scalene triangle $ABC$. The straight line through $A$ parallel to side $BC$ cuts the circumcircle of the triangle $ABC$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the lines perpendicular on $BC$, $CA$ and $AB$ from $A_1$, $B_1$ and $C_1$, respectively, are concurrent.
The quadrilateral $ABCD$ is inscribed on a circle of diameter $AC$ and such that the straight line $AC$ is the perpendicular bisector of $BD$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Choose points $F$ on the ray $DA$ and $G$ on the ray $BA$ so that $DG$ and $BF$ are parallel. The point $H$ is the orthogonal projection of $C$ on the line $FG$. Prove that points $B, E, F$ and $H$ are concyclic.
In the triangle $ABC$, let $D, E$ and $F$ be the midpoints of the sides and $P, Q$ and $R$ the midpoints of the medians
$AD$, $BE$ and $CF$, respectively. Prove the value of$$\frac{AQ^2 + AR^2 + BP^2 + BR^2 + CP^2 + CQ^2}{AB^2 + BC^2 + CA^2}$$does not depend on the shape of the triangle and find its value.
Consider a triangle $ABC$ with $\angle B > 90^o$ , such that for a point $H$ on the side AC, we have $AH = BH$ and $\angle HBC = 90^o$. Let $D$ and $E$ be the midpoints of $AB$ and $BC$, respectively. Through $H$, a parallel to $AB$ ́drawn, cutting $DE$ into $F$. Prove that $\angle BCF = \angle ACD$.
Let $\Gamma$ be the circumcircle of a triangle $ABC$. Let $ P$ be a point on $\Gamma$. Let $D$ and $E$ be the feet of the perpendiculars from $ P$ on the lines $AB$ and $BC$, respectively. Determine the locus of the circumcenter of the triangle $PDE$ while $P$ moves along $\Gamma$.
Let $ABC$ be an acute triangle. Let $AA_1$ and $BB_1$ be altitudes , with $A_1$ on $BC$ and $B_1$ on $AC$. Let $D$ be a point on the arc $AB$ of the circumcircle of $ABC$ that contains $C$. Lines $AA_1$ and $BD$ intersect at $ P$ and lines $BB_1$ and $AD$ intersect at $Q$. Prove that the midpoint of $PQ$ is on the line $A_1B_1$.
The circles $\omega_1$, with center $O_1$ and $\omega_2$, with center $O_2$, are tangent externally at point $T$. A third circle $\Omega$, with center $O$, ́is tangent to $\omega_1$ at $A$ and $\omega_2$ at $B$, and such that $O_1$ and $O_2$ are inside $\Omega$. the tangent line to $\omega_1$ and $\omega_2$ at $T$ cuts $\Omega$ at $K$ and $L$. Let $D$ be the midpoint of $KL$. Prove that $\angle O_1OO_2 = \angle ADB$.
In a triangle $ABC$, $I$ is the incenter and $O$ ́is the circumcenter. Choose point $P, Q$ and $R$ in segments $IA$, $IB$ and $IC$, respectively, such that$$IP \cdot IA = IQ \cdot IB = IR \cdot IC.$$Prove that points $I$ and $O$ lie on the Euler line of the triangle $PQR$.
The incircle of triangle $ABC$ has center $I$ and tangent sides $AB$ and $AC$ at points$ P$ and $Q$, respectively. Lines $BI$ and $CI$ intersect line $PQ$ at points $K$ and $L$, respectively. Prove that the circumcircle of the triangle $IKL$ is tangent to the incircle of triangle $ABC$ if and only if $AB + AC = 3BC$
Let $ABC$ be a triangle. Choose a $D$ point inside it. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ a circle that passes through $C$ and $D$ such that the other intersection point of $\omega_1$ and $\omega_2$ is on the line $AD$. Let $E$ and $F$ be the points in $BC$ that cut $\omega_1$ and $\omega_2$, respectively, with $E \ne B$ and $F \ne C$. Let $X$ a intersection of $DF$ and $AB$ and $Y$ the intersection of $DE$ and $AC$. Show that $XY$ is parallel to $BC$.
Let $ABCD$ be a convex quadrilateral that has an inscribed circle with center $I$. Suppose the straight line that passes through $A$, perpendicular on $AB$ cuts the line $BI$ at $M$ and that the line that passes through $A$, perpendicular on $AD$ cuts the line $DI$ at $N$. Prove that $MN$ ́is perpendicular on $AC$.
Let $O$ and $I$ be the circumcenter and incenter of the triangle $ABC$, respectively. Let $D$ be the touchpoint of the incircle of $ABC$ with $BC$. Let $E$ and $F$ be the intersections of the lines $AI$ and $AO$ with the circumcircle of $ABC$, respectively, with $A \ne E$ and $A \ne F$. Let $S$ be the intersection ̧of the lines $IF$ with $DE$, $M$ the intersection ̧of the lines $SC$ with $BE$ and $N$ the intersection of lines $AC$ with $BF$. Prove that the points $M, I$ and $N$ are collinear.
Let $AD$, $BE$ and $CF$ be the altitudes of the acute triangle $ABC$. Let $P, Q$ and $R$ be the feet of the perpendiculars from $A, B$ and $C$ on the lines $EF$, $FD$ and $DE$, respectively. Prove that the lines $AP$, $BQ$ and $CR$ are concurrent.
Let $ABC$ be an equilateral triangle. For a point $M$ inside the triangle $ABC$, let $D$, $E$ and $F$ be the feet of perpendiculars from $M$ on sides $BC$, $CA$ and $AB$, respectively. Find the locus of the points $M$ for which $\angle FDE = 90^o$.
Let ABCD be a circumscribed around a circle $\Gamma$ . Knowing that $\angle A = \angle B = 120^o$, $\angle D = 90^o$, and $BC=1$, determine the length of the side $AD$.
Let $ABC$ be an acute triangle. Denote by $D$ the foot of the perpendicular from $A$ on the side $BC$, by $M$ the midpoint of $BC$ and by $H$ the orthocenter of $ABC$. Let $E$ be the intersection point of the circle $\Gamma$ of the triangle $ABC$ with the ray $MH$ and let $F$ be the intersection (different from $E$) of the line $ED$ with the circle $\Gamma$. Prove that$$\frac{BF}{CF}=\frac{BA}{CA}.$$
Let $ABC$ be a triangle right at $A$ and let $I$ be its incenter. Line $BI$ intersects $AC$ at $D$, and line $CI$ intersects $AB$ at $E$. Let $ P$ and $Q$ be the feet of the perpendiculars from $D$ and $E$ on $BC$, respectively. Prove that $IP = IQ$.
Let $ABCD$ be a convex quadrilateral. Suppose that points $M$ and $N$ are on sides $AB$ and $BC$, respectively such that $[AMCD] = [CMB]$ and $[ANCD] = [ANB]$. Demonstrate that the line $MN$ passes through the midpoint of the diagonal $BD$ .
Note: $[P]$ denotes the area of the polygon $P$.
Let $ABC$ be a triangle and $H$ its orthocenter. The circumcircle of $ABC$ intersects the circle of diameter $AH$ at $P \ne A$. Prove that $HP$ passes through the midpoint of $BC$.
Let $ABC$ be a triangle and let $D, E, F$ be the touchpoints of the incircle of $ABC$ with sides $BC$, $CA$, $AB$, respectively. The segment $AD$ intersects the incircle of $ABC$ again at point $ P$. A perpendicular on $AD$ from $P$ intersects $EF$ at point $Q$. Lines $DE$ and $DF$ intersect line $AQ$ at $X$ and $Y$ , respectively. Show that $A$ is the midpoint of $XY$.
Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose the internal bisector of angle $\angle B$ cuts $AC$ at point $D$ and $BC = BD + AD$. Determine the measure of the angle $\angle A$.
Let $ABC$ be a triangle and let $M, N$ and $P$ be points on the line $BC$ such that $AM$, $AN$ and $AP$ are altitude, angle bisector and median of the triangle, respectively. It is known that$$\frac{[AMP]}{[ABC]}=\frac14 \,\,\,, \,\,\, \frac{[ANP]}{[ABC]}= 1 - \frac{\sqrt3}{2}.$$Determine the angles of triangle $ABC$.
Note: $[P]$ denotes the area of the polygon $P$.
Let $ABCD$ be a tetrahedron and let $E, F, G, H, K$ and $L$ be points on the segments $AB$, $BC$, $CA$, $DA$, $DB$ and $DC$, respectively, so that$$AE \cdot BE = BF\cdot CF = CG \cdot AG = DH \cdot AH = DK \cdot BK = DL \cdot CL.$$Prove that the six points marked on the sides of the tetrahedron are on the same sphere.
Let $ABC$ be a triangle with all its acute angles, of altitudes $AD$, $BE$, and $CF$ (with $D$ in $BC$, $E$ in $AC$ and $F$ in $AB$). Let $M$ be the midpoint of segment $BC$. The circle circumscribed around the triangle $AEF$ cuts line $AM$ at $A$ and $X$. Line $AM$ cuts line $CF$ at $Y$ . Let $Z$ be the intersection point between lines $AD$ and $BX$. Prove that the lines $YZ$ and $BC$ are parallel.
Let $ABC$ be an acute triangle such that its inscribed circle intersects the sides $AB$ and $AC$ at $D$ and $E$, respectively. Let $X$ and $Y$ be the respective intersection points of the internal angle bisectors of the angles $\angle ACB$ and $\angle ABC$ with the segment $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $XYZ$ is equilateral if, and only if, $\angle BAC = 60^o$.
Let $ABCD$ be a cyclic quadrilateral, and let $P$ be the intersection of the diagonals $AC$ and $BD$. Let $K$ and $L$ be the feet of the perpendiculars from $P$ on sides $AD$ and $BC$, respectively. If $M$ ́is the midpoint of $AB$, prove that $MK = ML$.
Let $ABC$ be an acute and scalene triangle, with $\angle BAC = 30^o$. The internal and external angle bisectors of $\angle ABC$ intersect the line $AC$ at $B_1$ and $B_2$, respectively. The internal and external angle bisectors of $\angle ACB$ intersect the line $AB$ at $C_1$ and $C_2$, respectively. Suppose the circles of diameters $B_1B_2$ and $C_1C_2$ intersect at a point $P$ inside the triangle $ABC$. Prove that $\angle BPC = 90^o$.
Let $ABC$ be an acute triangle, and let $P, Q, R$ be the midpoints of the arcs $BC,CA,AC$ of the circumcircle of $ABC$. Prove that $AP, BQ, CR$ intersected by $BC, CA, AB$ at $L, M, N$, respectively, prove that:$$\frac{AL}{PL}+\frac{BM}{QM} +\frac{LN}{RN} \ge 9$$
Given a triangle isosceles $ABC$ with $AB = BC$. A point $M$ is chosen within $ABC$ such that $\angle AMC = 2\angle ABC$. A point $K$ is on the segment $AM$ so that $\angle BKM = \angle ABC$. Prove that $BK = KM + MC$.
Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2\angle ADB$, $\angle ABD = 2\angle CDB$ and $AB = CB$. Prove that $AD = CD$.
Let $M$ be the intersection point of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. The internal bisector of the angle $\angle ACD$ intersects the ray $BA$ at a point $K$. It is known that $MA \cdot MC + MA \cdot CD = MB \cdot MD$. Prove that $\angle BKC = \angle CDB$.
Let $ABCD$ be an ́convex quadrilateral such that $\angle DAB = 90^o$. Let $M$ be the midpoint of $BC$. Knowing that $\angle ADC = \angle BAM$, prove that $\angle ADB = \angle CAM$.
On the circumcircle of triangle $ABC$, let $A_1$ be the point diametrically opposite vertex $A$. Let $A' $ be the intersection point to that of $AA_1$ and $BC$. The perpendicular to the line $AA'$ through $A'$ intersects sides $AB$ and $AC$ into $M$ and $N$, respectively. Prove that the points $A, M, A_1, N$ are on a circle whose center is at the altitude from vertex $A$ of triangle $ABC$.
Let $ABC$ be a triangle, $I$ its incenter and $D$ the foot of the perpendicular of$ I$ to the side $BC$. Let $P$ and $Q$ be the orthocenters of the triangles $AIB$ and $AIC$, respectively. Prove that $P, Q, D$ are collinear.
Let $ABC$ be a triangle with orthocenter $H$. The lines $AH$, $BH$, $CH$ intersect the circumcircle of $ABC$ again at $D, E, F$, respectively. Determine the maximum value of $\frac{Area\,(DEF)}{Area \,(ABC)}$.
Let $ABC$ be a sharp-angled triangle of circumcenter $O$ and let $D, E, F$ be the feet of altitudes on the sides $BC$, $CA$, $AB$, respectively. Let $\omega$ be the circumcircle of $ABC$, let $\omega'$ is the circle of $DEF$ and let $P$ be a variable point about $\omega$. Consider a circle tangent internally to $\omega$ at by $P$ and tangent externally the $\omega'$ at $Q$. Prove that the line $PQ$ passes through a fixed point on the line $HO$.
The point $P$ located internally in the triangle $ABC$ , such that $\angle BAP = \angle PAC = 20^o$, $\angle PCA = 10^o$ and $\angle PCB = 30^o$. Calculate the measure of the angle $\angle PBC$.
Let $H$ be the orthocenter and $M$ the midpoint of the side $BC$ in the acute-angled triangle $ABC$. Let $N$ be a point belonging to the extension of the side $HM$ such that $HM = MN$. Prove that $M$ belongs to circumscribed circle of triangle $ABC$.
Two given right triangles are such that the incircle of the first is exactly equal to the circumcircle of the second. Let $S$ and $S'$ be the areas of the first and second triangle, respectively. Prove that$$\frac{S}{S'}\ge 3+2\sqrt2$$
Let $ABCD$ be a convex quadrilateral. Let $n \ge 2$ be an integer. Prove that there are $n$ triangles of the same area with the following properties:
$\bullet$ There are no overlapping triangles
$\bullet$ Each triangle is contained within or within the perimeter of $ABCD$
$\bullet$ The sum of the areas of all triangles ́and at least $\frac{4n}{4n + 1}$ of the area of the quadrilateral $ABCD$.
Determine all angles of a ́convex quadrilateral $ABCD$ such that $\angle ABD = 29^o$, $\angle ADB = 41^o$, $\angle ACB =82^o$ and $\angle ACD = 58^o$.
In a convex quadrilateral $ABCD$, the angles $\angle A$ and $\angle C$ have the same measure and the bisector of $\angle B$ passes through the midpoint of the side $CD$ . If $CD = 3AD$, determine the ratio $\frac{AB}{BC}$
Let $ABC$ be a triangle. Construct isosceles triangles $BCD$, $CAE$ and $ABF$ externally to $ABC$, with base $BC$, $CA$ and $AB$, respectively. Prove that the lines that pass through $A, B$ and $C$ and are perpendicular to $EF$, $FD$ and $DE$, respectively, are concurrent.
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$ with $BC > CA$. Let $F$ be the foot of the altitude from $C$ of this triangle. The line perpendicular to $OF$ at point $F$ intersects the line $AC$ at point $P$. Prove that $\angle FHP = \angle BAC$.
The $ABCD$ quadrilateral is inscribed in a circle. Let $M$ be the intersection point of their diagonals and $L$ the midpoint of the arc $AD$ that does not contain other vertices of the quadrilateral. Prove that the distances from $L$ to centers of the triangles inscribed in the triangles $\vartriangle ABM$ and $\vartriangle CDM$ are equal.
In the figure below, $ABC$ is an isosceles triangle with $BA = BC$. Point D is inside it so that $\angle ABD = 13^o$, $\angle ADB = 150^o$ and $\angle ACD = 30^o$. Furthermore, $ADE$ is an equilateral triangle. Determine the value of the angle $\angle DBC$.
Squares $BAXX'$ and $CAYY'$ are built externally on the sides of the isosceles triangle $ABC$ with equal sides $AB = AC$. Let $E$ and $F$ be the feet of the perpendiculars of an arbitrary point $K$ of the segment $BC$ on segments B$Y$ and $CX$ , respectively. Let $D$ be the midpoint of $BC$.
a) Prove that $DE = DF$.
b) Find the locus of the midpoint of $EF$
Clipping a convex $n$-gon means choosing a pair of consecutive sides $AB$, $BC$ and replacing them by three segments $AM$, $MN$ and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, we cut the $MBN$ triangle to obtain an convex $(n + 1)$-gon. A regular hexagon $P_6$ of Area $ 1$ is clipped to get a heptagon $P_7$ . So $P_7$ is clipped in one of the $7$ possible ways to get an octagon $P_8 $, and so on. Prove that no matter how clippings happen, the area of $P_n$ is greater than $\frac13$ for all $n \ge 6$.
The equilateral triangle $DCE$ is constructed externally on the side $DC$ of the parallelogram $ABCD$ . Let $X$ be an arbitrary point in the plane. Prove that $XA + XB + AD \ge XE$.
Let $P$ be the intersection point of the diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $AB = AC = BD$. Let $O$ and $I$ be the circumcenter and incenter of the triangle $ABP$. Prove that if $O\ne I$, then it is $OI \perp CD$.
Let $ABC$ be a triangle with $AC = BC$. A point $P$ internal to $ABC$ is such that $\angle PAB = \angle PBC$. Le $M$ be the midpoint of $AB$, prove that $\angle APM + \angle BPC = 180^o$.
Let $P =\{P_1, P_2, ..., P_{1997}\}$ be a set of 1997 points within a circle of radius $ 1$, with $P_1$ being the center of the circle. For $k = 1, 2, ..., 1997$ let $x_k$ be the distance from $P_k$ to the point of $P$ closest to $P_k$. Prove that $x_1^2+x_2^2+...+x_{1997}^2 \le 9$.
Let $ABCDE$ be a convex pentagon. Suppose $BD \cap CE = A'$, $CE \cap DA = B'$, $DA \cap EB = C'$, $EB \cap AC = D'$ and $AC \cap BD = E'$. Also suppose that $(ABD') \cap (AC'E) = A''$, $ (ECB') \cap (BD'A) = B''$, $(CDA') \cap (CE'B) = C''$, $(DEB') \cap DA'C = D''$ and $(EAC') \cap (EB'D) = E''$. Prove that $AA''$, $BB''$ , $CC''$, $DD''$ and $EE''$ are concurrent.
Let $D$ be a point on the side $BC$ of the triangle \vartriangle ABC such that $AD =\frac{BD^2}{AB + AD} =\frac{CD^2}{AC+AD}$. Let E be a point such that $D$ lies on $[AE]$ and $CD =\frac{DE^2}{CD+CE}$. Prove that $AE = AB + AC$.
Given the $\vartriangle ABC$ isosceles with $AB = AC > BC$. The perpendicular bisector of $AB$ meets the external bisector of $\angle ADB$ at point $P$, The perpendicular bisector of $AC$ meets the external bisector of $\angle ADC$ at point $Q$. Prove that $B, C, P$ and $Q$ are concyclic.
Given a convex polygon $A_1A_2...A_n$ of n sides. Prove or disprove that the largest circles circumscribed to the triangles $A_iA_jA_k,$ with $ i, j, k \in \{1, 2, ..., n\}$ and $i\ne k \ne j$, are of the form $A_{\ell}A_{\ell+1}A_{\ell+2}$.
Let $D$ be a point on side $AB$ of a triangle $ABC$. A point $L$ lies inside triangle $ABC$ such that $BD = LD$ and $\angle LAB = \angle LCA = \angle DCB$. We know that $\angle ALD + \angle ABC = 180^o$. Prove that $\angle BLC = 90^o$.
Let $I$ be the incenter and $AB$ the shortest side of a triangle $ABC$. The circle with center $I$ passing through $C$ intersects the ray $AB$ into $P$ and the ray $BA$ into $Q$. Let $D$ be the touchpoint between the circle exscribed to $ABC$, relative to point $A$ and side $BC$. Let $E$ be the the reflection point of $C$ wrt to point $D$. Prove that $PE \perp CQ$.
Let $G$ be the centroid of a right triangle $ABC$ with $\angle BCA = 90^o$. Let P be a point on the ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on the ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circles circumscribed around the triangles $AQG$ and $BPG$ intersect at a point on the side $AB$.
Let $ABC$ be an acute triangle. Points $M$ and $N$ are the midpoints of sides $AB$ and $BC$, respectively, and $BH$ is the altitude relative to $AC$, with $H$ over $AC$. The circles circumscribed around the triangles $AHN$ and $CHM$ intersect again at point $P$, $P \ne H$. Prove that PH passes through the midpoint of $MN$.
Let $ABC$ be an acute triangle and $O$ its circumcenter. Point $H$ is the foot of the altitude relative to vertex $A$ and the points $P$ and $Q$ are the feet of the perpendiculars of $H$ to sides $AB$ and $AC$, respectively. Given that $AH^2 = 2AO^2$, prove that $O, P$, and $Q$ are collinear.
A parallelogram $ABCD$ is given, with AB < AC < BC. Points $E$ and $F$ are selected on the circle $\omega$ of $ABC$ such that the tangent to $\omega$ at these points pass through $D$ and segments $AD$ and $CE$ intersect. Knowing that $\angle ABF = \angle DCE$, determine the angle $\angle ABC$.
Let $ABC$ be a triangle with $\angle BAC =\frac{\pi}{6}$ and circumradius $ 1$. For a point $X$ inside or on the edge of the triangle, let $m(X) = \min \, (AX, BX, CX)$. determine the angles of ABC if $\max \, (m(X)) = \frac{\sqrt3}{3}$.
Let $ABCD$ be a trapezoid with $AB \parallel CD$. The sides $DA$, $AB$, $BC$ are tangent to a circle touching $AB$ in $P$. The sides $BC$, $CD$, $DA$ are tangent to a circle that plays $CD$ in $Q$. Prove that the lines $AC$, $BD$ and $PQ$ are concurrent.
The convex quadrilateral $ABCD$ satisfies $\angle B = \angle C$ and $\angle D = 90^o$. Suppose $|AB| = 2|CD|$. Prove that the bisector of $\angle ACB$ is perpendicular to $CD$.
Let $\vartriangle ABC$ be a triangle such that $AC = BC$. $P$ is a point on arc $AB$ of the circumcircle of $ABC$, that does not contain $C$. Let $D$ be the foot of the perpendicular from $C$ to $PB$. Prove that $PA + PB = 2PD$.
Let $\vartriangle ABC$ be an obtuse-angled triangle in $C$ such that $2\angle BAC = \angle ABC$. Let $P$ be a point on the side$ AB$ such that $BP = 2BC$. Let $M$ be the midpoint of $AB$ ($M$ is between $P$ and $B$). Show that the perpendicular on $AC$ through of $M$ intersects $PC$ at its midpoint.
In a given triangle $\vartriangle ABC$, $\angle A = 90^o$ and $M$ is the midpoint of $BC$. Choose $D$ in segment $AC$ such that $AM = AD$ and let $P$ be the other point of intersection between the circumcircles of triangles $\vartriangle AMC$ and $\vartriangle BDC$. Prove that $P$ lies on the bisector of angle $\angle ACB$.
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