geometry shortlists from IberoAmerican Mathematical Olympiads (OIM)
with aops links in the names
Olimpíada Iberoamericana de Matemática (OIM)
geometry shortlists collected inside aops:
1992-95, 2001-04, 2008, 2010-19
complete so far
Let $ABC$ be a triangle of circumcenter $O$ and incenter $I$, and let $A_1$ be the touchpoint of the inscribed circle with $BC$. Lines $AO$ and $AI$ intersect the circumscribed circle again at $A'$ and $A''$, respectively. Show that $A'I$ and $A''A_1$ intersect at a point on the circle circumscribed around $ABC$.
Let $A$ and $B$ be two points outside the circle $K$. If we draw the segment $ \overline{AB}$, it intersects the circle at points $M$ and $N$ ($M \in \overline{AN}$). We take a point $C$ inside the circle and draw the segments $ \overline{AC}$ and $ \overline{BC}$, so that $ \overline{AC}$ intersects the circle at $Q$ and $ \overline{BC}$ at $P$. If $M$ and $Q$ are equidistant from $A$ and $N$, and $P$ from $B$ and there is a point $R$ interior to triangle $ABC$ such that$$\angle RMN = \angle RNM = \angle RPC = \angle RQC = 30^o.$$Prove that the area of said circle is less than or equal to the area of the circle circumscribed to triangle $ABC$.
Let $H$ be the orthocenter of a triangle inscribed in the circle $K$ and, $P$ and $Q$ any two points of said circle and $M$ a point of $HP$ such that $\frac{x}{u}+\frac{y}{v}$ is minimal, $u$ and $v$ being the distances from $M$ on $PQ$ and $HQ$ respectively, $x$ is the distance from $H$ on $PQ$ and $y$ the distance from $P$ on $HQ$. Determine the locus of the points $M$.
Show that if in an isosceles triangle $ABC$, with equal sides $AB$ and $AC$, it is true that$$\frac{AC}{CB}=\frac{1+\sqrt5}{2}=\phi$$then angle $A$ measures $36^o$ and the other two $72^o$ each.
If we have two balls, $B$ and $R$, in a circular billiard of $ 1$ m radius, located on the same diameter at $0.8$ and $0.5$ m. from the center, in which points of the band should one of them affect so that, in the first rebound, it hits the other?
(It is assumed that we throw the ball without effect and that there are no frictions).
Let $A, B, C$ be three points of a line $\ell$ and $M$ a point outside $\ell$. Let $A '$, $B'$, $C '$ be the centers of the circles circumscribed to the triangles $MBC$, $MCA$ , $MAB$ respectively.
a) Prove that the projections of $M$ on the sides of triangle $A'B'C'$ are collinear.
b) Prove that the circle circumscribed around the triangle $A'B'C'$ passes through $M$.
Let the triangle $ABC$ such that $BC = 2 $and $AB> AC$. We draw the altitude $AH$, the median $AM$ and the angle bisector $AI$. Given are $MI = 2- \sqrt3$ and $MH = \frac12$.
a) Calculate $AB^2-AC^2$.
b) Calculate the lengths of sides $AB$, $AC$, and the median $AM$.
c) Calculate the angles of triangle $ABC$.
Let $ABCD$ be a parallelogram and $MNPQ$ be a square inscribed in it with $M \in AB$, $N \in BC$, $P \in CD$, and $Q \in DA$.
a) Prove that the two quadrilaterals have the same center $O$.
b) What rotation sends $M$ to $N$?
c) Construct the square $MNPQ$ given the parallelogram $ABCD$.
Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.
In a triangle $ABC$, points $A_{1}$ and $A_{2}$ are chosen in the prolongations beyond $A$ of segments $AB$ and $AC$, such that $AA_{1}=AA_{2}=BC$. Define analogously points $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$. If $[ABC]$ denotes the area of triangle $ABC$, show that $[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]$.
Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle.
a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$, $B$, $C$ is 5.
b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$, $BP$, $CP$. Also, the area of such triangle is $\frac{\sqrt{3}}{4}$.
Let $ABCD$ be a convex and cyclic quadrilateral. Let $M$ be a point in $DC$ such that the perimeter $ADM$ is equal to the perimeter of quadrilateral $AECM$ and that the area of triangle ADM is equal to the area of quadrilateral $ABCM$. Prove that the quadrilateral $ABCD$ has equal sides.
Let $ABC$ be an equilateral triangle and $\Gamma$ its incircle. If $D$ and $E$ are points on the segments $AB$ and $AC$ such that $DE$ is tangent to $\Gamma$, show that $\frac{AD}{DB}+\frac{AE}{EC}=1$.
Let $ABCD$ be a square with side $2a$. Draw a circle of diameter $AB$ and another circle of radius $a$ and center $B$ . Denote by $I$ the intersection of the last one with $AB$, and by $E , G$ the intersections of the other circles, $E$ being inside the square and $G$ outside. Finally consider the circle inscribed in the square of center $O$ and let $F$ be the point of intersection of this with the arc $OB$. Let $K$ be the midpoint of $CD$. Prove that $FK$ is parallel to $BE$.
a) Let $OABC$ be a tetrahedron such that $OA = OB = OC = 1$ and $\angle AOB = \angle BOC = \angle COA$ (*). Determine the maximum volume of all these tetrahedra.
b) Consider all the regular pyramids $OA_1...OA_n$ such that $OA_1 = OA_2 = ... = OA_n = \ell$ and $\angle A_1OA_2 = \angle A_2OA_3 = ... = \angle A_nOA_1$ (*) .Determine the radius of the circle circumscribed to the polygon $A_1A_2. ..A_n$ where $A_1A_2 ... A_n$ is the base of the pyramid $OA_1A_2 ... A_n$ which has the maximum volume.
(*) This condition can be removed.
Construct a triangle $ABC$ if the points $M, N$ and $ P$ in $AB, AC$ and $BC$ respectively , are known and verify the property$$\frac{MA}{MB} = \frac{PB}{PC} = \frac{NC}{CA} = k$$where $k$ is a fixed number between $-1$ and $0$.
Outside a triangle $ABC$ the equilateral triangles $ABM$, $BCN$ and $CAP$ are constructed. Given the acute triangle $MNP$ , construct the triangle $ABC$.
Show that for every convex polygon whose area is less than or equal to $1$, there exists a parallelogram with area $2$ containing the polygon.
Let $P_1P_2P_3P_4P_5$ be a convex pentagon in the plane. Let $Q_i$ be the point of intersection of the segments that join the midpoints of the opposite sides of the quadrilateral $P_ {i + 1} P_ {i + 2} P_ {i + 3} P_ {i + 4}$ where $P_ {k + 5 } = P_k$ with $k \in N$, $i \in \{1,2,3,4,5 \}$. Prove that the pentagons $P_1P_2P_3P_4P_5$ and $Q_1Q_2Q_3Q_4Q_5$ are similar.
Let $ A$ and $ B$ be set in the plane and define the sum of $A$ and $B$ as follows: $A \oplus B = \{a + b | a \in A$ and $B \in B \}$.
Let $T_1, T_2, .., T_n$ be equilateral triangles with side $\ell$ in the plane and let $S$ equal the area of $T_1 \oplus T_2 \oplus ... \oplus T_n$. Prove that
$$\sqrt3 \cdot 2 ^ {2n-2} \cdot \ell ^ 2 \le S \le \frac {3 \cdot 2 ^ {n-2} \cdot \ell ^ 2} {\tan (\pi / (2 ^ n \cdot 3))}$$
Let $A,\ B$ and $C$ be given points on a circumference $K$ such that the triangle $\triangle{ABC}$ is acute. Let $P$ be a point in the interior of $K$. $X,\ Y$ and $Z$ be the other intersection of $AP, BP$ and $CP$ with the circumference. Determine the position of $P$ such that $\triangle{XYZ}$ is equilateral.
$ABC$ is any triangle. Equilateral triangles $PAC$, $CBQ$ and $ABR$ are constructed, with $P$, $Q$ and $R$ outside $ABC$. $G_1$ is the center of $PAC$, $G_2$ is the center of $CBQ$, and $G_3$ is the center of $ABR$. If area $(PCA) +$ area $(CBQ)$ + area $(ABR)$= $K$ (constant), find the maximum area of triangle $G_1G_2 G_3$.
A draftsman constructs a sequence of circles in the plane as follows:
The first circle has its center at $(0,0)$ and radius $1$ and for all $n\ge 2$ the $n$-th circle has a radius equal to one-third of the radius of the $(n-1)$-th circle and passes through its center. The only point that belongs to all these circles is painted red. Determine the locus of the red dots in all possible constructions.
Let $ A$ and $ B$ be two points on a circle $\Gamma$. The tangents at $ A$ and $ B$ meet at $P$. A secant is constructed through $P$ that cuts $\Gamma$ at $C$ and $D$. Prove that that the tangent at $C$, the tangent at $D$, and the line $AB$ are concurrent.
The six edges of a tetrahedron are given. Knowing that $ a$ and $a'$, $ b$ and $b'$, $c$ and $c'$ are opposite edges construct, with a ruler and compass, the height of the tetrahedron relative to the face $(abc)$.
The rays $OX$, $OY$ and $OZ$ are given such that $\angle XOY = \angle YOZ = \alpha$. A line segments intersects those rays at $A, B$ and $C$ ($B$ between $A$ and $C$) determining the segments $OA = a$, $OB = b$, $OC = c$. Prove that$$\frac{a+c}{b} \ge \frac{2}{\cos \alpha}.$
Let $M_a$, $M_b$ and $M_c$ be the midpoints of the sides of a triangle $ABC$, and $H_a$, $H_b$ and $H_c$ the feet of the altitudes of the triangle with vertices $M_a$, $M_b$ and $M_c$. Prove that the centers of the circles $(ABC)$, $(M_aM_bM_c)$ , $(H_aH_bH_c)$ and the circle inscribed to the triangle $H_aH_bH_c$ are collinear.
Notation: The circle $(ABC)$ denotes the circle that passes through $A$, $B$ and $C$.
Let $ ABCD$ a quadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on $ AB$, that
is tangent to the other three sides of the quadrilateral.
(i) Show that $ AB = AD + BC$.
(ii) Calculate, in term of $ x = AB$ and $ y = CD$, the maximal area that can be reached for such quadrilateral.
Consider a regular polygon with $n$ sides. On two consecutive sides $AB$ and $BC$, equilateral triangles $ABD$ and $BCE $are built externally. Determine the values of $n$ so that it can be ensured that there exists a regular polygon with consecutive sides $DB$ and $BE$.
Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$.
Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.
In an acute triangle $ABC$, the distance from the orthocenter to the centroid is half the radius of the circumscribed circle. $D, E$, and $F$ are the feet of the altitudes of triangle $ABC$. If $r_1$ and $r_2$ are, respectively, the radii of the inscribed and circumscribed circles of the triangle $DEF$, determine $\frac{r_1}{r_2}$ ,.
Let $P$ be a convex polygon of area $A$ and perimeter $p$. The isoperimetric ratio of $P$ is defined as $I(P) =\frac{4 \pi A}{p^2}$. The roundness of $P$ is defined as $R(P) =\frac{r_1}{r_2}$, where $r_1$ is the maximum radius for a circle inside $P$, and $r_2$ the minimum radius of a circle containing $P$. Prove that $R(P) <I(P)$.
Two circles intersect at points $P$ and $Q$, and the distance between their centers is $d$. Starting from a point $A$, variable on one of them, draw the lines $AP$ and $AQ$ that intersect the other circle at $ B$ and $C$.
a) Prove that the radius of the circumference circumscribed to triangle $ABC$ equals $d$.
b) Determine the locus of the center of the circumscribed circle of the triangle $ABC$ when $A$ moves along the first circle.
In a triangle $ABC$, $D$, $E$ and $F$ are points on the sides $BC$, $CA$ and $AB$ respectively, so that the lines $AD$, $BE$ and $CF$ would intersect at a point $P$. If $R$ is a point on the segment $AP$, show that $E, R$ and $F$ are collinear, if and only if
$$\frac{AP}{PD} \cdot \frac{PR}{AR}= \frac{FP}{FC} + \frac{EP}{EB}$$
The incircle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. Let the line $AD$ intersect this incircle of triangle $ABC$ at a point $X$ (apart from $D$). Assume that this point $X$ is the midpoint of the segment $AD$, this means, $AX = XD$. Let the line $BX$ meet the incircle of triangle $ABC$ at a point $Y$ (apart from $X$), and let the line $CX$ meet the incircle of triangle $ABC$ at a point $Z$ (apart from $X$). Show that $EY = FZ$.
Let $ABC$ be a triangle right at $C$. Let $m$ and $n$ be the lengths of the medians from $A$ and $B$ respectively.
(a) Prove that $\frac12 \le \frac{m}{n} \le 2$.
(b) If we call the angle formed by those medians $\alpha$, find the maximum value of $\alpha$.
let $P_1$ be a regular polygon with $ r$ sides, and $P_2$ be a regular polygon with $s$ sides, whose interior angles are $a$ and $ b$ respectively. If $\frac{a}{b}=\frac{59}{58}$. Find the smallest and largest possible value of $ r$.
The incircle of the triangle $\triangle{ABC}$ has center at $O$ and it is tangent to the sides $BC$, $AC$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. The lines $BO$ and $CO$ intersect the line $YZ$ at the points $P$ and $Q$, respectively.
Show that if the segments $XP$ and $XQ$ has the same length, then the triangle $\triangle ABC$ is isosceles.
Given a circle with center $O$ and radius $ r$, a straight line $s$ and a point $A$, we consider a point $P$ variable in $s$. The circle of center $ P$ and radius $PA$ cuts the initial circle at $B$ and $C$. Find the locus of the midpoint $Q$ of chord $BC$ as $ P$ varies in $s$.
Let $ABC$ be a triangle such that angle $A$ is $60^o$. Let $P$, $Q$ be the feet of the perpendiculars drawn from $ A$ on the internal bisectors of angles $B$ and $C$ respectively. Given that $BP = 104$ and $CQ = 105$, find the perimeter of triangle $ABC$.
Let $H$ be the intersection point of the altitudes $AD$, $BE$ and $CF$ of an acute triangle $ABC$ and let $A'$, $B'$, $C'$ be the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Suppose that $H$ does not lie into the interior of triangle $A'B'C'$. Show that at least one of the areas of triangles $AEF$, $BDF$, and $CDE$ is less than $1/9$ of the area of triangle $ABC$.
Let $M,N$ be arbitrary points on sides $AC$ and $BC$ of triangle $ABC$, respectively, and $ P$ an arbitrary point of segment $MN$. Show that at least one of the triangles $AMP$ and $BNP$ has an area less than or equal to $1/8$ of the area of triangle $ABC$.
Let $P$ be a point in the interior of the equilateral triangle $\triangle ABC$ such that $\sphericalangle{APC}=120^\circ$. Let $M$ be the intersection of $CP$ with $AB$, and $N$ the intersection of $AP$ and $BC$. Find the locus of the circumcentre of the triangle $MBN$ as $P$ varies.
In a triangle $\triangle{ABC}$ with all its sides of different length, $D$ is on the side $AC$, such that $BD$ is the angle bisector of $\angle{ABC}$. Let $E$ and $F$, respectively, be the feet of the perpendicular drawn from $A$ and $C$ to the line $BD$ . Let $M$ any point the point on $BC$ . Prove that $DM \perp BC$ iff $\angle{EMD}=\angle{DMF}$.
Let $h_a$, $h_b$ and $h_c$, respectively, be the lengths of the altitudes corresponding to the vertices $A$, $B$ and $C$ of a triangle $ABC$. Show that if it is verified that $h_a = h_b + h_c$ then the line determined by the feet of the interior bisectors of angles $B$ and $C$ passes through the centroid of the triangle.
An acute triangle $ABC$ and two points $T$ and $M$ are considered, on sides $BC$ and $AC$, respectively. Circles $\Gamma$ and $\Gamma '$ of diameters $AT$ and $BM$ are drawn, respectively. Let $P$ and $Q$ be the points of intersection of these circles. Show that $P$ and $Q$ are collinear with the orthocenter $H$ of triangle $ABC$.
In triangle $ABC$ the bisector of angle $A$ intersects $BC$ at $D$. Show that if $BD$ is equal to the radius of the circle circumscribed by $ABC$, then the following are verified:
a) $(ADC) =\frac{b^2}{4}$
b) The measure of angle $C$ is strictly greater than $30^o$ and strictly less than $150^o$.
Let $\Gamma_1$ be a circle of diameter $AB$, $C$ any point of $\Gamma_1$, different from $A$ and different from $B$, $D$ the projection of $C$ on $AB$, $\Gamma_2$ the circle of center $A$ and radius $AD$. Let $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$, and let $R$ be the intersection point of $AC$ and $PQ$. Prove that $\frac{PR}{RQ}=\frac13$ if and only if $\angle RDQ=90^o$
We consider two circumferences $C$ and $C'$ of centers $O$ and $O'$, respectively, secant at $ A$ and $ B$ such that $OA\perp O'A$ and $OB \perp O'B$. Let $I$ be the midpoint of the segment $OO'$. A line $r \perp AI$ is drawn through $ A$, which cuts again $C$ at $M$ and $ C'$ at $N$. Show that $ A$ is the midpoint of $MN$.
Points $ A$, $ B$ and $C$ are given, in this order, on a line $t$, and $D$ is the midpoint of $BC$. On the same side of $t$, draw the semicircles of diameters $AB$ and $AC$. The perpendicular drawn by $D$ on line $t$ intersects the major semicircle at $P$. Line $DT$ is tangent to the minor semicircle at $T$.
a) Show that $DP = DT$.
b) Show that points $ A$, $T$, and $P$ are collinear.
An angle $\angle XOY = \alpha$ and points $ A$ and $ B$ on $OY$ are given such that $OA = a$ and $OB = b$ ($a> b$). A circle passes through points $ A$ and $ B$ and is tangent to $OX$.
a) Calculate the radius of this circle in terms of $a, b$ and $\alpha$.
b) If $a$ and $b$ are constant and $\alpha$ varies, show that the minimum value of the radius of the circumference is $\frac{a-b}{2}$
Let $P$ be a point on the arc $AB$ of the circumscribed circle of the square $ABCD$. Segments $AC$ and $PD$ intersect at $Q$ . Segments $AB$ and $PC$ intersect at $R$. Show that $QR$ is the bisector of angle $\angle PQB$.
Let $\ell_1$, $\ell_2$ be two parallel lines and $\ell_3$ a line perpendicular to $\ell_1$, and to $\ell_2$ at $H$ and $P$, respectively. Points $Q$ and $R$ lie on $\ell_1$ such that $QR = PR$ ($Q\ne H$). Let $d$ be the diameter of the circle inscribed in the triangle $PQR$. Point $T$ lies on $\ell_2$ in the same half plane as $Q$ wrt the line $\ell_3$ such that:$$\frac{1}{TH}=\frac{1}{d}-\frac{1}{PH}$$Let $X$ be the intersection point of $PQ$ and $TH$. Find the locus of points $X$ as point $Q$ moves along $\ell_1$ with the exception of $H$.
Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.
Let $O$ be the circumcenter of the isosceles triangle $ABC$ with $AB = AC$. Let $P$ be a point on the segment $AO$ and $Q$ be the symmetric of $P$ wrt the midpoint of $AB$. If $OQ$ intersects $AB$ at $K$ and the circle through $A$, $K$ and $O$ intersects $AC$ at $L$, prove that $\angle ALP = \angle CLO$.
In a square $ABCD$, let $P$ and $Q$ be points on the sides $BC$ and $CD$ respectively, different from its endpoints, such that $BP=CQ$. Consider points $X$ and $Y$ such that $X\neq Y$, in the segments $AP$ and $AQ$ respectively. Show that, for every $X$ and $Y$ chosen, there exists a triangle whose sides have lengths $BX$, $XY$ and $DY$.
A regular polygon of $2004$ vertices, $A_1 A_2, ..., A_{2004}$ is given.
Prove that the lines$$A_2A_{1005}, \,\,\, A_{670}A_{671},\,\,\, A_{1338}A_{1340}$$are concurrent and geometrically characterize the point of intersection.
Given a scalene triangle $ ABC$. Let $ A'$, $ B'$, $ C'$ be the points where the internal bisectors of the angles $ CAB$, $ ABC$, $ BCA$ meet the sides $ BC$, $ CA$, $ AB$, respectively. Let the line $ BC$ meet the perpendicular bisector of $ AA'$ at $ A''$. Let the line $ CA$ meet the perpendicular bisector of $ BB'$ at $ B'$. Let the line $ AB$ meet the perpendicular bisector of $ CC'$ at $ C''$. Prove that $ A''$, $ B''$ and $ C''$ are collinear.
Let $D$ be the foot of the interior bisector of angle $A$ in triangle $ABC$. The line joining the centers of the circles inscribed in $ABD$ and $ACD$ intersects$ AB$ at $M$ and AC at $N$. Prove that $BN$ and $CM$ intersect on the angle bisector $AD$.
In the plane are given a circle with center $ O$ and radius $ r$ and a point $ A$ outside the circle. For any point $ M$ on the circle, let $ N$ be the diametrically opposite point. Find the locus of the circumcenter of triangle $ AMN$ when $ M$ describes the circle.
In triangle $ABC$, $P$ and $Q$ are points on side $BC$ such that lines $AQ$ and $AP$ form equal angles with sides $AB$ and $AC$, respectively. Let $BT_1$ and $CT_2$ be the tangents to the circle circumscribed to $APQ$ drawn from $B$ and $C$. If $M$ is the point where the interior angle bisector from $A$ (in triangle $ABC$) intersects side $BC$, prove that$$\left(\frac{BT_1}{CT_2} \right)^2 = \frac{AB}{AC} \cdot \frac{MB}{MC}$$
In triangle $ABC$, let $M \in AB$, $N \in AC$, $ P = MN \cap BC$, $Q = CM \cap BN$, $R = AQ \cap BC$. Finally, let $k = \frac{PB}{PC}$. Show that the necessary and sufficient condition for the centroid of $ABC$, $G$, to lie on $MN$, is that
$$\frac{AQ}{RQ}=\frac{(2k+1)^2}{k(k+1)}$$
Given a triangle $ ABC$, let $ r$ be the external bisector of $ \angle ABC$. $ P$ and $ Q$ are the feet of the perpendiculars from $ A$ and $ C$ to $ r$. If $ CP \cap BA = M$ and $ AQ \cap BC=N$, show that $ MN$, $ r$ and $ AC$ concur.
A trapezoid $ABCD$ with $AB$ parallel to $CD$ and $AD <CD$ is inscribed in a circle $\Gamma$. Let $DP$ a chord parallel to $AC$. The tangent line to $\Gamma$ that passes through $D$ intersects the line $AB$ at $E$ . Lines $PB$ and $DC$ intersect at $Q$. Show that $EQ = AC$.
Let $ABCD$ be a parallelogram of area $ 1$. The points $P, Q, R$ and $S$ belong to the sides $AB$, $BC$, $CD$ and $DA$, respectively. Prove that one of the triangles $PBQ$, $QCR$, $RDS$, $SAP$ has area less than or equal to $1/8$.
Let $ABC$ be a triangle and let $P$ be a point belonging to the angle bisector $AD$, with $D$ on the side $BC$. Let $E, F, G$ be the second intersections of $AP, BP, CP$ with the circumcircle of the triangle, respectively. Let $H$ be the intersection of $EF$ and $AC$, and $I$ the intersection of $EG$ and $AB$. Let $M$ be the midpoint of $BC$.
Prove that the lines $AM$, $BH$, and $CI$ have a point in common.
Let $ABC$ be a triangle and let $P$ be a point inside the triangle. Let $E, F, G$ be the second intersections of $AP$, $BP$, $CP$ with the circumcircle of the triangle, respectively. Let $H$ be the intersection of $EF$ and $AC$, and $I$ the intersection of $EG$ and $AB$. Let $P_a$ be the intersection of $BH$ and $CI$. We define Pb and Pc similarly. Show that $AP_a$, $BP_b$ and $CP_c$ have a point in common.
Let $ABC$ be an acute triangle with $AB <BC$. Let $BH$ be an altitude with $H$ on $AC$, $I$ the incenter of triangle $ABC$ and $M$ the midpoint of $AC$. Line $MI$ intersects $BH$ at point $N$. Prove that $BN <IM$.
Let $ABC$ be a triangle of inradius $ r$ and let $D, E, F$ be the tangencies of the ex-circles of $ABC$ with sides $BC$, $CA$, $AB$, respectively. If $P$ is the intersection between $AD$, $BE$, $CF$ and $x, y, z$ are the distances from $P$ to lines $BC$, $CA$, $AB$, show that$$\frac{x + y + z}{3} \ge r$$
Let $ ABC$ a triangle and $ X$, $ Y$ and $ Z$ points at the segments $ BC$, $ AC$ and $ AB$, respectively.Let $ A'$, $ B'$ and $ C'$ the circuncenters of triangles $ AZY$,$ BXZ$,$ CYX$, respectively.Prove that $ 4(A'B'C')\geq(ABC)$ with equality if and only if $ AA'$, $ BB'$ and $ CC'$ are concurrents.
Note: $ (XYZ)$ denotes the area of $ XYZ$
We say that a convex polygon $P$ can be doubled into a convex polygon $Q$ when a paper with the format of the polygon $P$ it can be folded so that it covers the surface of $Q$ exactly two times. The folds are all straight. Prove that if $P$ can be doubled into $Q$ then $Q$ cannot be a regular polygon with five or more sides.
Let $A'$, $B'$ and $C'$ be points on the sides $BC$, $CA$ and $AB$ of triangle $ABC$ respectively, such that$$\frac{BA'}{A'C}=\frac{CB'}{B'A}=\frac{AC'}{C'B}=2$$. The circle that passes through points $A'$, $B'$ and $C'$ cuts again to the sides $BC$, $CA$ and $AB$ at $A''$, $B''$ and $C''$ respectively. Let $A_1$, $B_1$ and $C_1$ be points on $B''C''$, $C''A''$ and $A''B''$ such that$$\frac{B''A_1}{A_1C''}=\frac{C''B_1}{B_1A''}=\frac{A''C_1}{C_1B''}=2.$$Prove that lines $AA_1$, $BB_1$ and $CC_1$ are concurrent.
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear
Let $ABC$ be an acute triangle and $H$ its orthocenter. Lines $BH$ and $CH$ intersect $AC$ and $AB$ at $D$ and $E$, respectively. The circumcircle of $ADE$ intersects the circumcircle of $ABC$ at $F\ne A$. Prove that the internal bisectors of $\angle BFC$ and $\angle BHC$ intersect at a point on segment $BC$.
Let $ABC$ be an acute triangle and $H$ its orthocenter. Lines $BH$ and $CH$ intersect $AC$ and $AB$ in $D$ and $E$, respectively. Line $DE$ intersects line $BC$ at $P$. Let $\Gamma$ be the circle that passes through points $A$, $D$ and $E$, and $\Gamma'$ the circle that passes through $ B$, $H$ and $C$. Line $AP$ cuts $\Gamma$ again at $I$. Let $J$, on segment $BC$, be the intersection of the internal bisector of $\angle BHC$ and $BC$ and $M$ the midpoint of arc $BC$ of $\Gamma'$ containing $H$. Line $MJ$ intersects $G'$ again at $N$. Prove that the triangles $DIE$ and $CNB$ are similar.
Two circles $C_1$ and $C_2$, with centers $O_1$ and $O_2$ respectively, intersect at two points $ A$ and $ B$. Let $X$ and $Y$ be points on $C_1$, different from $A$ and $B$. Lines $XA$ and $YA$ intersect $C_2$ again at $Z$ and $W$ respectively.
Let $M$ be the midpoint of $O1O2$, S the midpoint of $XA$, and $T$ the midpoint of $WA$. Prove that $MS = MT$ if and only if $XYZW$ is cyclic.
Let $ABC$ be an acute triangle in which the altitudes $AA_1$, $BB_1$, and $CC_1$ have been drawn. Let $A_2$ be a point of the segment $AA_1$ such that $\angle BA_2C =90^o$. Similarly are defined points $B_2$ and $C_2$. Let $A_3$ be the point of intersection of segments $B_2C$ and $BC_2$. Similarly are defined points $B_3$ and $C_3$. Prove that segments $A_2A_3$, $B_2B_3$, and $C2C3$ are concurrent
The circle $ \Gamma $ is inscribed to the scalene triangle $ABC$. $ \Gamma $ is tangent to the sides $BC, CA$ and $AB$ at $D, E$ and $F$ respectively. The line $EF$ intersects the line $BC$ at $G$. The circle of diameter $GD$ intersects $ \Gamma $ in $R$ ($ R\neq D $). Let $P$, $Q$ ($ P\neq R , Q\neq R $) be the intersections of $ \Gamma $ with $BR$ and $CR$, respectively. The lines $BQ$ and $CP$ intersects at $X$. The circumcircle of $CDE$ meets $QR$ at $M$, and the circumcircle of $BDF$ meet $PR$ at $N$. Prove that $PM$, $QN$ and $RX$ are concurrent.
Let $S_1, S_2,…, S_n$ be a family of equilateral triangles, all with parallel sides and of the same orientation. For each triangle If $T_i$ is its medial triangle. We define finally $S$ as the union of all $S_i$ and $T$ triangles as the union of all $T_i$ triangles. Prove what$$area \,\,\, S\le 4 area \,\,\, T$$
Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.
In triangle $ABC$, let $D, E$ and $F$ be the midpoints of$ BC$, $CA$ and $AB$, respectively. Show that the triangle $ABC$ is similar to the one formed by its medians $AD$, $BE$, $CF$ if and only if one of the sides $BC$, $CA$, $AB$ is a common tangent to two of the three circles circumscribed to $GAB$, $GBC$, $GCA$, where $G$ the centroid of $ABC$.
Let $ABC$ be a triangle. Let $D$ on side $BC$ such that the measure of angle $DAC$ is twice the measure of angle $BAD$. Let $I$ be the center of the circle $\Gamma$ inscribed in the triangle $ADC$. The circle circumscribed to the triangle $AIB$ intersects $\Gamma$ at $X$ and $Y$. Let $P$ be the intersection of $XY$ with $AI$. Let $M$ be the foot of the perpendicular from $I$ to $AB$. Prove that $4AP \cdot PI = MI^2$.
Let $ABC$ be a triangle and $P$ a point in its interior. Consider the circle $\Gamma_A$ that passes through $P$, $B$ and $C$, and let $A'$ be the second intersection of the line $AP$ with $\Gamma_A$. In an analogous way $B'$ and $C'$ are constructed. Determine the possible values of
$$\frac{A'B}{A'C} \cdot \frac{B'C}{B'A} \cdot\frac{C'A}{C'B} $$
We say that a nondegenerate quadrilateral is inscribed in an equilateral triangle if the $4$ vertices of the quadrilateral are contained in the lines that form the sides of the triangle and if in each of these $3$ lines there is at least one vertex of the quadrilateral. Determine the maximum and minimum number of equilateral triangles in which a convex quadrilateral may be inscribed.
Let $ABC$ be a triangle and $X,Y,Z$ be the tangency points of its inscribed circle with the sides $BC, CA, AB$, respectively. Suppose that $C_1, C_2, C_3$ are circle with chords $YZ, ZX, XY$, respectively, such that $C_1$ and $C_2$ intersect on the line $CZ$ and that $C_1$ and $C_3$ intersect on the line $BY$. Suppose that $C_1$ intersects the chords $XY$ and $ZX$ at $J$ and $M$, respectively; that $C_2$ intersects the chords $YZ$ and $XY$ at $L$ and $I$, respectively; and that $C_3$ intersects the chords $YZ$ and $ZX$ at $K$ and $N$, respectively. Show that $I, J, K, L, M, N$ lie on the same circle.
Given the triangle $ABC$, let $\Gamma_C$ and $\Gamma_B$ be the tangent circles in $A$ to the sides $AC$ and $AB$ that pass through $B$ and $C$ respectively. $\Gamma_C$ and $\Gamma_B$ again intersect the line $BC$ at $P$ and $Q$, respectively. Let $A'$, $B'$, $C'$ be the midpoints of $PQ$, $CA$ and $AB$, respectively. Let $R$ be the intersection of $PB'$ and $QC'$. prove that the circles circumscribed to the triangles $C'AB'$, $A'BC'$ and $B'CA'$ intersect at $R$.
Let $I$ be the incenter of triangle $ABC$ and suppose that the perpendicular lines drawn from $I$ on $IA, IB, IC$ intersect a given tangent line of the incircle of triangle $ABC$ at points $P, Q, R$ respectively. Prove that $AP$, $BQ$ and $CR$ concur.
Let $ABCD$ be a rectangle. Construct equilateral triangles $BCX$ and $DCY$, in such a way that both of these triangles share some of their interior points with some interior points of the rectangle. Line $AX$ intersects line $CD$ on $P$, and line $AY$ intersects line $BC$ on $Q$. Prove that triangle $APQ$ is equilateral.
In a triangle $ABC$ right at $ B$, let $I$ and $O$ be it's incenter and circumcenter respectively. If $\angle AIO= 90^o$ and the area of triangle $AIC$ is $\frac{10060}{7}u^2$. What is the area of quadrilateral $ABCI$?
In a triangle $ABC$, the sides $a, b$ and the interior angle bisector $b_c = 4$ are known. If the area of the triangle is equal to $a + b$. Determine the angle $\angle C$.
Given a convex quadrilateral $ABCD$, with $CD = a$, $BC = b$, $\angle DAB = 90^o$, $\angle DCB = \alpha$, $AB = AD$, find the length of the diagonal $AC$.
In a right triangle, the hypotenuse is equal to$ \sqrt6+\sqrt2$ and the radius of the circle inscribed to the triangle equal to $\frac{\sqrt2}{2}$ . Determine the acute angles of the right triangle.
Let $ABC$ be an acute triangle, $D$ and $E$ be the feet of the altitudes drawn from $B$ and $C$, respectively, and $H$ its orthocenter. Points $M$ and $N$ are chosen on segments $AB$ and $AC$, respectively, such that $MN$ is parallel to $BC$. Let $P$ be the intersection of $BN$ with $CM$, and $Q$ the intersection of the perpendicular bisectors of $MN$ and $DE$. Show that $P, Q$, and $H$ are collinear.
In a convex quadrilateral $ABDC$, $\angle CBD = 10^o$, $\angle CAD=20^o$, $\angle ABD=40^o$, $\angle BAC = 50^o$. Determine the measure of the angles $\angle BCD$ and $\angle ADC$.
There is a family of circles $S_1,..,S_m$ forming a figure $S$, and a family of equilateral triangles $T_1,...,T_n$ with parallel sides and of the same orientation forming a figure $T$. It is known that for every triangle $T_i$ we have that area $S\cap T_i \,\,\, \ge \lambda $ area $T_i$. Prove that: area $T \le \frac{13}{\lambda}$ area $S$.
Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI = AR$.
Let $ABC$ be an acute triangle of orthocenter $H$, let $M$ be the midpoint of $BC$, and let $\omega$ be the circumcircle of triangle $ABC$. Let $D$, $E$, $F$, $G$ be the points of intersection of the rays $AH$, $BH$, $CH$, $MH$ with $\omega$, respectively. Let $I, J$ be the incenters of $GDF$ and $GDE$ respectively. Show that $IJ$ is parallel to $BC$.
Let $\vartriangle ABC$ be an acute triangle inscribed in a circle $\omega$ and let $L,M,P$ be the midpoints of segment $AB$, of segment $AC$ and minor arc $BC$ of $\omega$, respectively. Let $X$ and $Y$ be points in $\omega$ such that $\angle XLB= \angle BLP$ and that $\angle YMC= \angle CMP$.
a) Prove how that the line joining the centers of the circumscribed circles of the triangles $\vartriangle XLP$ and $\vartriangle YMP$, is parallel to $BC$.
b) If $XP$ and $AB$ intersect at $U$, and $YP$ and $AC$ intersect at $V$, prove that $UV$ is parallel to $BC$.
Given an acute scalene triangle $ABC$, let $ P$ and $Q$ be the feet of the perpendiculars drawn from $ A$ and $ B$, respectively, and $D$ be any point on the circumcircle of triangle $ABC$. Let $R$ be the intersection of segments $BD$ and $AP$, and $S$ the intersection of segments $AD$ and $BQ$. Find the locus of the circumcenter of the triangle $RSD$ as point $D$ varies.
Let $\Gamma$ be a circunference and $O$ its center. $AE$ is a diameter of $\Gamma$ and $B$ the midpoint of one of the arcs $AE$ of $\Gamma$. The point $D \ne E$ in on the segment $OE$. The point $C$ is such that the quadrilateral $ABCD$ is a parallelogram, with $AB$ parallel to $CD$ and $BC$ parallel to $AD$. The lines $EB$ and $CD$ meets at point $F$. The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at $I$.
Prove that the line $EI$ is the angle bissector of $\angle BEC$.
Let $ABC$ and $ADE$ be two triangles whose altitudes from $A$ have the same length, whose circumscribed circles are externally tangent and also $\angle BAC= \angle DAE$.
$\bullet$ Prove that that lines $BE$ and $CD$ are parallel.
$\bullet$ Prove that lines $BD$ and $CE$ intersect on the common tangent to the circles circumscribed , passing through point A.
Let $X$ and $Y$ be the diameter's extremes of a circunference $\Gamma$ and $N$ be the midpoint of one of the arcs $XY$ of $\Gamma$. Let $A$ and $B$ be two points on the segment $XY$. The lines $NA$ and $NB$ cuts $\Gamma$ again in $C$ and $D$, respectively. The tangents to $\Gamma$ at $C$ and at $D$ meets in $P$. Let $M$ the the intersection point between $XY$ and $NP$. Prove that $M$ is the midpoint of the segment $AB$.
The trapezoid $ABCD$ is considered, with $AB \parallel CD$, $AB> CD$, circumscribed to the circle $k(O,r)$ of center $O$ and radius $r$. If $d_1=|AC|$ and $d_2=|BD|$, prove that$$d_1^2+d_2^2 \ge 16r^2$$
Let $ABCD$ be a parallelogram with an obtuse angle at $A$. Let $P$ be a point on segment $BD$ so that the circle with center at $P$, passing through $A$, intersects line $AD$ at A and $Y$ and intersects line $AB$ at $A$ and $X$. Line $AP$ intersects line $BC$ at $Q$ and line $CD$ at $R$, respectively. Prove that $\angle XPY=\angle XQY +\angle XRY.$
Let $ABCD$ be a convex quadrilateral and let $ P$ be the intersection of diagonals $AC$ and $BD$. The radii of the circles inscribed in the triangles $ABP$, $BCP$,$CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.
Let $ABC$ be a non-isosceles triangle, of incenter $I$, centroid $G$ and sides $a,b$ and $c$ (opposite to side of the vertices $A, B$ and $C$ respectively). Show that the necessary and sufficient condition for the line $IG$ to be perpendicular to $BC$ is that $b + c = 3a$.
Let $ABC$ be an acute triangle and $H$ its orthocenter. Let $D$ be the intersection of the altitude from $A$ to $BC$. Let $M$ and $N$ be the midpoints of $BH$ and $CH$, respectively. Let the lines $DM$ and $DN$ intersect $AB$ and $AC$ at points $X$ and $Y$ respectively. If $P$ is the intersection of $XY$ with $BH$ and $Q$ the intersection of $XY$ with $CH$, show that $H, P, D, Q$ lie on a circumference.
Let $ABC$ be an acute triangle and $D$ be the foot of altitude from on $BC$. Let $E$ and $F$ be the midpoints of $BD$ and $DC$, respectively. Let $O$ and $Q$ be the circumcenters of the triangles $ABF$ and $ACE$, respectively. Let $ P$ be the intersection of $OE$ and $QF$. Prove that $PB=PC$.
Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points of the plane such that $C$ is interior to segment $AM$ and $D$ is interior to segment $BN$. Suppose that the lines $NA$ and $NC$ cut the lines $MB, MD$ at points $E,F, G$ and $H$. Show that points $E,F, G$, and $H$ lie on a circle if and only if $ABCD$ is a rhombus.
We will say that four circles form an Olympian flower if they are centered on the vertices of a cyclic quadrilateral and every two contiguous circles are tangent. We will call these touchpoints as tangency counting points. Let $ABCD$ be a quadrilateral with right angles at $B$ and $D$. Show that there exists exactly one set of four circles that form an Olympian flower whose tangency counting points are precisely $A,B,C$ and $D$ and calculate the radii of the circles in terms of the side lengths of the quadrilateral.
$ABCD$ is a convex quadrilateral inscribed in a circle with center $O$. On the sides $AB$ and $CD$ are considered points $F$ and $E$. respectively, such that $EO=FO$. The lines $AD$ and $BC$ intersect the line $EF$ at the points $M$ and $N$, respectively. Finally, the point $P$ is symmetric to $M$ wrt the midpoint of the segment $AE$. Prove that the triangles $FBN$ and $CEP$ are similar.
Let $ABC$ be a triangle such that $AB> AC$, with a circumcircle $\omega$. The tangents to $\omega$ are drawn from $ B$ and $C$. and these intersect at $ P$. The perpendicular on $AP$ through $A$ intersects $BC$ at $R$. Let S be a point on segment $PR$ such that $PS=PC$.
a) Prove that lines $CS$ and $AR$ intersect on $\omega$.
b) Let $M$ be the midpoint of $BC$ and $Q$ the point of intersection of $CS$ and $AR$. If $\omega$ and the circumcircle of $AMP$ intersect at a point $J$ ($J\ne A$), prove that $P,J$ and $Q$ are collinear.
Let $A_1A_2A_3$ be a non-isosceles triangle with circumcircle $\Omega$ and orthocenter $H$. Circles with diameters $A_iH$ intersect with $\Omega$ at points $B_i$ ($\ne A_i$). Lines $A_i B_{i+1}$ and $A_{i+1}B_i$ intersect at $C_i$. Show that the points $C_i$ and the centroid of $A_1A_2A_3$ are collinear. (The subscripts $i + 1$ are taken modulo $3$).
Let $ABC$ be a triangle and $ P$ an interior point. Let points $A_1$, $B_1$ and $C_1$ be the reflections of $A,B$ and $C$ wrt point $ P$. respectively. Let points $A'$,$B'$ and $C'$ be the reflections of $A,B$ and $C$ wrt $B_1C_1$,$C_1A_1$ and $A_1B_1$ respectively. Prove that the triangles $A'B'C'$ and $ABC$ are similar.
Let $ABC$ and $ADE$ be two right isosceles triangles that share only point $A$, which is the vertex opposite the hypotenuse for both triangles. The order in which the vertices are given is that of clockwise, for both triangles. Let $M, N, P$ and $Q$ be the midpoints of the segments $BE$, $CD$, $BC$ and $DE$, respectively. Show that the segments $MN$ and PQ intersect and are perpendicular.
Let $ABC$ be an acute triangle and let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $P$ be a point on segment $AD$. Lines $BP$ and $CP$ intersect sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $J$ and $K$ be the feet of the perpendiculars from $E$ and $F$ to $AD$, respectively. Show that $\frac{FK}{KD}=\frac{EJ}{JD}$.
Let $\Gamma_1$ be a circle with center $O, T$ be a point interior to $\Gamma_1$ and $\Gamma_2$ be the circle of diameter $OT$. Construct, using a ruler and compass, a chord $PQ$ of $\Gamma_1$, parallel to $OT$ and such that $PT$ and $QO$ intersect at a point $M$ of $\Gamma_2$.
In the circle circumscribed to the triangle $ABC$, the respective midpoints $K, L, N$ are considered of arcs $BC$, $CA$ and $AB$ that do not contain the third vertex. Lines $NK$ and $KL$ intersect BC at points $A_1$ and $A_2$, respectively. $KL$ and $NL$ intersect $CA$ at $B_1$ and $B_2$,respectively. $LN$ and $NK$ intersect$ AB$ at $C_1$ and $C_2$, respectively. Prove that the proportions are verified
$$\frac{CA_1}{a}=\frac{CB_2}{b}=\frac{a + b}{a + b + c},$$$$\frac{CBC_1}{c}=\frac{BA_2}{a}=\frac{a + c}{a + b + c},$$$$\frac{AB_1}{b}=\frac{AC_2}{c}=\frac{b + c}{a + b + c},$$and that lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ intersect at the center of the circle inscribed in triangle $ABC$.
Note: As usual $a, b$ and $c$ are the lengths of the sides $BC, CA$ and $AB$ of the triangle, respectively.
A line $r$ contains the points $A$, $B$, $C$, $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$. Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that $\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$
Let $ABC$ be a triangle, $I$ its incenter and $X, Y, Z$ the touchpoints of the incircle with sides $BC$, $AC$, and $AB$, respectively. Let $P$ be the intersection of segment $YZ$ with line $IX$. Prove that $A, P$, and the midpoint of $BC$ are collinear.
Let $S$ be a plane convex polygon with the property that any triangle contained in $S$ has a perimeter less than or equal to $ 1$. Suppose that $S$ is inscribed in an equilateral triangle $\tau$. Show that the perimeter of $\tau$ is less than $2\sqrt3$
Let $B$ and $C$ be two fixed points and $\Gamma$ a fixed circle such that line $BC$ has no point of intersection with $\Gamma$. Point $A$ varies over $\Gamma$ so that $AB\ne AC$. Let $H$ be the orthocenter of triangle $ABC$. Let$ X \ne H$ be the second point of intersection of the circumcircle of the triangle $BHC$ and the circle of diameter $AH$. Find the locus of point $X$ when $A$ varies over $\Gamma$ .
Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $A_1$ be the midpoint of arc $BC$ containing $A$ and $A_2$ be the touchpoint of the incircle of $ABC$ with $BC$. Line $A_1A_2$ intersects $\Gamma$ at$ A_3$. Similarly, are defined $B_3$ and $C_3$ . Let us take the points $P, Q$ and $R$ so that $QR, PR$ and $PQ$ they are tangent to $\Gamma$ at $A_3$, $B_3$ and $C_3$ respectively. Prove that $PA$, $QB$, and $RC$ concur.
Let $ABCD$ be a trapezoid of bases $AD$ and $BC$, with $AD> BC$, whose diagonals intersect at point $E$. Let $ P$ and $Q$ be the feet of the perpendiculars drawn from $E$ to the sides $AD$ and $BC$, respectively, with $P\in AD$ and $Q \in BC$. Let $I$ be the incenter of triangle $AED$ and let $K$ be the point of intersection of the straight lines $AI$ and $CD$. If $AP + AE = BQ + BE$, prove that $AI= IK$.
Let $\omega_1$ and $\omega_2$ be two congruent circles that intersect at two different points $P$ and $Q$. $AC$ and $BD$ are secants of $\omega_1$ and $\omega_2$, respectively, such that their intersection point is $P$. $M$ and $N$ are defined as the midpoints of segments $BD$ and $AC$, respectively. The lines AM and BN intersect at $R$. Show that $RQ\perp MN$.
Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$. Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$, and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$. Prove that $L, J, A, Q,$ and $K$ all lie on a circle.
Circles $\Phi_1$ and $\Phi_2$ intersect at two different points $A$ and $B$. The tangent to $\Phi_1$ through $A$ intersects $\Phi_2$ at $M$ and the tangent to $\Phi_2$ through $A$ that intersects $\Phi_1$ at $N$. Let $P$ be the reflection of $A$ wrt $B$. Let $S$ and $T$ be the intersection points of the lines $PM$ and $PN$ with $\Phi_1$ and $\Phi_2$, respectively. Prove that the points $S, B, T$ are collinear.
Let $\omega_1$ and $\omega_2$ be internal tangent circles at a point $A$, with $\omega_2$ inside $\omega_1$. Let $BC$ be a chord of $\omega_1$, so it is tangent to $\omega_2$. Prove that the ratio between the length of $BC$ and the perimeter of triangle $ABC$ is constant, that is, it does not depend on the choice of chord $BC$ that is chosen to construct the triangle.
The circumferences $C_1$ and $C_2$ cut each other at different points $A$ and $K$. The common tangent to $C_1$ and $C_2$ nearer to $K$ touches $C_1$ at $B$ and $C_2$ at $C$. Let $P$ be the foot of the perpendicular from $B$ to $AC$, and let $Q$ be the foot of the perpendicular from $C$ to $AB$. If $E$ and $F$ are the symmetric points of $K$ with respect to the lines $PQ$ and $BC$, respectively, prove that $A, E$ and $F$ are collinear.
Prove that in any triangle$$\min \,\, (a, b, c) + 2 \max \,\,(m_a, m_b, m_c)\ge \max \,\,(a, b, c) + 2 \min (m_a, m_b, m_c),$$where $m_a, m_b, m_c$ denote the lengths of the medians, and $a, b, c$ denote the lengths of the sides.
In the plane, a polygon $P$, closed without auto intersections, is said to be an Ariel polygon when there exists a point $P$ such that every line passing through $P$ divides polygon P into two polygonal lines of the same length. Find all the polygons of Ariel.
Let $ABC$ be an acute triangle with $AC > AB$ and $O$ its circumcenter. Let $D$ be a point on segment $BC$ such that $O$ lies inside triangle $ADC$ and $\angle DAO + \angle ADB = \angle ADC$. Let $P$ and $Q$ be the circumcenters of triangles $ABD$ and $ACD$ respectively, and let $M$ be the intersection of lines $BP$ and $CQ$. Show that lines $AM, PQ$ and $BC$ are concurrent.
Let $A_1A_2A_3$ be a triangle. Let $h_i,w_i, m_i$ be the altitude, the angle bisector and the median, respectively, which start from vertex $A_i$. Prove that if $h_1$,$w_2$ and $m_3$ are concurrent, and also $h_2$,$w_3$ and $m_1$ are concurrent, then $A_1A_2A_3$ is equilateral.
Let $ABC$ be a triangle and let $D, E$ and $F$ be the points where the incircle of $ABC$ is tangent to sides $BC$, $CA$, and $AB$, respectively. We call $G, H$ and $I$ the points diametrically opposite to $D, E$ and $F$ respectively, wrt the incircle of $ABC$. Let us denote by $P, Q$ and $R$ the intersection points of the lines $BI$ with $CH$, the lines $AI$ with $CG$ and lines $AH$ with $BG$ respectively. Show that the lines $AP$, $BQ$ and $CR$ are concurrent.
Let $ABC$ be an acute triangle with $AC> AB$, circumcircle $\Gamma$ and $M$ midpoint of side $BC$. A point $N$ interior to $ABC$ is chosen, so that if $D$ and $E$ are its feet from the perpendicular on $AB$ and $AC$, respectively, then $DE \perp AM$. The circumcircle of $ADE$ intersects $\Gamma$ at $L$ ($L \ne A$) and lines $AL$ and $DE$ intersect at $K$. Line $AN$ intersects $\Gamma$ at $F$ ($F \in A$). Show that if $N$ is the midpoint of segment $AF$, then $KA = KF$.
Let $ABC$ be an acute angled triangle and $\Gamma$ its circumcircle. Led $D$ be a point on segment $BC$, different from $B$ and $C$, and let $M$ be the midpoint of $AD$. The line perpendicular to $AB$ that passes through $D$ intersects $AB$ in $E$ and $\Gamma$ in $F$, with point $D$ between $E$ and $F$. Lines $FC$ and $EM$ intersect at point $X$. If $\angle DAE = \angle AFE$, show that line $AX$ is tangent to $\Gamma$.
Let $ABC$ be a triangle; a circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. Let $P$ be the point of intersection of $BE$ and $CD$, the inner angle bisector of $\angle EPC$ intersects $AB$ and $AC$ at $F$ and $G$ respectively. Let $M$ be the midpoint of $FG$, the parallelograms $BMGW$ and $CMFY$ are constructed. If $Z$ is the intersection of $WG$ and $FY$ and $AP$ intersects the circumcircle of triangle $ ABC$ at $X \ne A$, show that quadrilateral $XYZW$ it is cyclic.
A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length $1$, two have length $2$ and two have length $3$. Show that $r$ satisfies the equation $2r^3 - 7r - 3 = 0$.
Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$ and $AB = AC$. Let $M$ be the midpoint of $BC$. A point $D \neq A$ is chosen on the semicircle with diameter $BC$ that contains $A$. The circumcircle of triangle $DAM$ cuts lines $DB$ and $DC$ at $E$ and $F$ respectively. Show that $BE = CF$.
Let $ABC$ be a triangle, $I$ its incenter, $\Gamma$ its circumscribed circle, and $M$ the midpoint of the side $BC$. Suppose there are two different points $P, Q$ on the line $IM$ such that $PB = PI$ and $QC = QI$. Let $G$ be the point of intersection of the lines $PB$ and $QC$. Prove that the line $IG$ and the perpendicular bisector of segment $BC$ intersect at a point of $\Gamma$.
$ABC$ is a triangle with incenter $I$. We construct points $P$ and $Q$ such that $AB$ is the bisector of $\angle IAP$, $AC$ is the bisector of $\angle QAI$, and $\angle PBI + \angle IAB = \angle QCI + \angle IAC = 90^o$. Lines $IP$, $IQ$ intersect $AB$, $AC$ at $K, L$, respectively. The circumscribed circles of the triangles $APK$ and $AQL$ intersect at $A$ and $R$. Show that $R$ lies the line joining the midpoints of $BC$ and $KL$.
Let $ABC$ be an acute triangle with $AB \ne AC$ and $H$ its orthocenter. Let $D$ and $E$ be the intersections of $BH$ and $CH$ with $AC$ and $AB$ respectively, and $ P$ the foot of the perpendicular from $A$ on $DE$. The circumcircle of $BPC$ intersects $DE$ at a point $Q \ne P$. Show that the lines $AP$ and $QH$ intersect at a point on the circumcircle of $ABC$.
Let ABC be an acute triangle with circumcenter $O$ and orthocenter $H$. The circle of center $X_A$ passing through points $A$ and $H$ is tangent to the circle circumscribed to $ABC$. $X_B$ and $X_C$ are analogously defined. Let $O_A$, $O_B$ and $O_C$ be the reflections of the point $O$ on the lines $BC$, $CA$ and $AB$, respectively. Show that the lines $O_AX_A$, $O_BX_B$ and $O_CX_C$ are concurrent.
In the triangle $ABC$, the inscribed circle $\omega$ is tangent to sides $BC$, $CA$ and $AB$ at points $D, E$ and $F$, respectively. Let $M$ and $N$ be the midpoints of $DE$ and $DF$, respectively. Points $D', E', F'$ are considered on the line $MN$ such that $D'E = D'F$, $BE' \parallel DF$ and $CF' \parallel DE$. Prove that the lines $DD'$, $EE'$ and $FF'$ are concurrent.
Let $ABC$ be an acute triangle with $AC > AB > BC$. The perpendicular bisectors of $AC$ and $AB$ cut line $BC$ at $D$ and $E$ respectively. Let $P$ and $Q$ be points on lines $AC$ and $AB$ respectively, both different from $A$, such that $AB = BP$ and $AC = CQ$, and let $K$ be the intersection of lines $EP$ and $DQ$. Let $M$ be the midpoint of $BC$. Show that $\angle DKA = \angle EKM$.
Let $ABC$ a triangle. The perpendicular bisector of the segment $AC$ cuts the line $AB$ at $P$ and the perpendicular bisector of the segment $AB$ cuts the line $AC$ at $Q$. The circle with center $P$ and radius $PA$ intersects the circle with center $Q$ and radius $QA$ at a point $X$ different from $A$. Prove that points $A$, $O$, and $X$ are collinear, where $O$ is the circumcenter of triangle $ABC$.
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and inscribed in a circumference $\Gamma$. Let $P$ and $Q$ be two points on segment $AB$ ($A$, $P$, $Q$, $B$ appear in that order and are distinct) such that $AP=QB$. Let $E$ and $F$ be the second intersection points of lines $CP$ and $CQ$ with $\Gamma$, respectively. Lines $AB$ and $EF$ intersect at $G$. Prove that line $DG$ is tangent to $\Gamma$.
Let $ABC$ be a triangle with $AC> A B$ and center $I$. The midpoints of the sides $BC$ and $AC$ are $M$ and $N$, respectively. If the lines $AI$ and $IN$ are perpendicular, prove that $AI$ is tangent to the circumcircle of the triangle $IMC$.
Let $ABC$ be a triangle. Let $D$ be the foot of altitude from $B$ on $AC$, and $E$ be the foot of altitude from $C$ on $AB$. Let $P$ be the point such that $AEPD$ is a parallelogram. Denote by $K$ the second intersection point of the circumscribed circles of the triangles $EBP$ and $DCP$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $BC$. It is known that the points $M$, $K$ and $N$ are collinear. Calculate the ratio $\frac{BC}{AB}$
Let $\Gamma$ be the circumcircle of triangle $ABC$. The line parallel to $AC$ passing through $B$ meets $\Gamma$ at $D$ ($D\neq B$), and the line parallel to $AB$ passing through $C$ intersects $\Gamma$ to $E$ ($E\neq C$). Lines $AB$ and $CD$ meet at $P$, and lines $AC$ and $BE$ meet at $Q$. Let $M$ be the midpoint of $DE$. Line $AM$ meets $\Gamma$ at $Y$ ($Y\neq A$) and line $PQ$ at $J$. Line $PQ$ intersects the circumcircle of triangle $BCJ$ at $Z$ ($Z\neq J$). If lines $BQ$ and $CP$ meet each other at $X$, show that $X$ lies on the line $YZ$.
Let $ABC$ is a triangle with circumcircle $\Gamma$. The circumcircle $C_b$ is inside the angle \angle CBA in the manner that $BA$ and $BC$ are tangents to $C_b$ and $C_b$ is externally tangent to $\Gamma$ at a point $ P$. In a similar way, define $C_c$ which is tangent to $\Gamma$ at $Q$. Let $P'$ and $Q'$ be the touch points of $C_c$ and $C_b$ with $BC$, respectively. Prove that $PP'$ and $QQ'$ intersect on the angle bisector of the $\angle BAC$.
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