geometry shortlists from Baltic Way with aops links in the names
collected inside aops:
2009-2020
missing 2012, 2018
Assume that triangle $ABC$ is not equilateral and that both $\beta = \angle ABC$ and $\gamma = \angle ACB$ are larger than $30^o$. Let $O$ be the orthocentre of triangle $ABC$. Let the triangles $ACB'$ and $ABC'$ be equilateral with $ B$ and $B'$ on opposite sides of $AC$ and $C$ and $C'$ on opposite sides of $AB$. Let $B''$ and $C''$ be such interior points of the segments $BB'$ and $CC'$ that $BB'' =\frac12 \left(1- \frac{\tan (90^o-\beta)}{\tan 60^o}\right) BB''$ and $CC'' =\frac12 \left(1- \frac{\tan (90^o-\gamma)}{\tan 60^o}\right) CC''$.
Prove $\angle B''OC'' = 120^o$.
Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$, and let $K$ be a point on the ray $BA$ beyond $A$. The line $KM$ intersects the side $BC$ at the point $L$. $P$is the point on the segment $BM$ such that $PM$ is the bisector of the angle $LPK$. The line $\ell$ passes through $A$ and is parallel to $BM$. Prove that the projection of the point $M$ onto the line $\ell$ belongs to the line $PK$.
In a quadrilateral $ABCD$ we have $AB||CD$ and $AB=2CD$. A line $\ell$ is perpendicular to $CD$ and contains the point $C$. The circle with centre $D$ and radius $DA$ intersects the line $\ell$ at points $P$ and $Q$. Prove that $AP\perp BQ$.
The point $H$ is the orthocentre of a triangle $ABC$, and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point.
The triangle $ABC$ is isosceles with $AB = AC$. The point $ P$ inside $ABC$ satisfies two conditions:
(i) $A$ lies on the trisector line of $\angle BPC$, i.e. $AP$ meets $BC$ at $Q$ such that $\angle BPC = 3 \angle QPC$
(ii) $\angle BPQ = \angle BAC$.
Show that $Q$ trisects $BC$, i.e. $BC = 3 QC$.
Let $AB$ be the diameter of the circle $\Gamma$ with centre $O$ and let $C$ and $D$ be points on $\Gamma$, on different sides on $AB$ and such that $AD$ and $CB$ intersect at $R$. The circumscribed circles of the triangles $AOC$ and $BOD$ meet also at $Q$. $CD$ and $AB$ meet at $P$. Show that $Q, P$ and $R$ are collinear.
Six circular mint cookies, each of radius greater than $1$, are given. Show that it is impossible to place them all upon a circular plate of radius $3$ without overlaps.
For which $n\ge 2$ is it possible to find $n$ pairwise non-similar triangles $A_1, A_2,\ldots , A_n$ such that each of them can be divided into $n$ pairwise non-similar triangles, each of them similar to one of $A_1,A_2 ,\ldots ,A_n$?
A unit square is cut into $m$ quadrilaterals $Q_1,\ldots ,Q_m$. For each $i=1,\ldots ,m$ let $S_i$ be the sum of the squares of the four sides of $Q_i$. Prove that
\[S_1+\ldots +S_m\ge 4\]
In a rectangle $ABCD$ where $AB = 2BC$, the diagonals intersect in a point$ E$, the angle bisector of the angle $\angle CAD$ intersects the side $CD$ in a point $F$ and the diagonal $BD$ in a point $G$, and $EG= 25$. Determine the length of $FC$.
Let $ABCD$ be a convex quadrilateral with precisely one pair of parallel sides.
$(a)$ Show that the lengths of its sides $AB,BC,CD, DA$ (in this order) do not form an arithmetic progression.
$(b)$ Show that there is such a quadrilateral for which the lengths of its sides $AB ,BC,CD,DA$ form an arithmetic progression after the order of the lengths is changed.
Does there exist a non-equilateral triangle, such that the angle between any two of its medians equals $120^o$?
Assume that all angles of a triangle $ABC$ are acute. Let $D$ and $E$ be points on the sides $AC$ and $BC$ of the triangle such that $A, B, D,$ and $E$ lie on the same circle. Further suppose the circle through $D,E,$ and $C$ intersects the side $AB$ in two points $X$ and $Y$. Show that the midpoint of $XY$ is the foot of the altitude from $C$ to $AB$.
Let $ABCD$ be a square and let $S$ be the point of intersection of its diagonals $AC$ and $BD$. Two circles $k,k'$ go through $A,C$ and $B,D$; respectively. Furthermore, $k$ and $k'$ intersect in exactly two different points $P$ and $Q$. Prove that $S$ lies on $PQ$.
A lizard wants to walk from one corner to the diametrically opposite corner of a regular dodecahedron with edge length $ 1$. Prove that the lizard has to walk a distance of at least $4$.
(A regular dodecahedron is a Platonic solid consisting of twelve regular pentagons.)
Given a circle such that it is possible to fit inside it six circles with radius $ r$ so that they do not overlap. Prove that it is also possible to fit inside it seven circles with radius $ r$ so that they do not overlap.
The circles $C_1$ and $C_2$ intersect at $A$ and $B$. The points $P$ and $Q$ are on $C_2$, $ P$ in the interior and $Q$ in the exterior of $C_1$. The lines $AP$ and $BP$ meet $C_1$ also at $X$ and $Y$, respectively, while the lines $QA$ and $QB$ meet $C_1$ also at $Z$ and $T$. Show that $X Y = Z T$ .
Let $AD$, $BE$ and $CF$ be the angle bisectors of triangle $ABC$. Assume$$\frac{1}{AE}+\frac{1}{AF}=\left(\frac{1}{\sqrt{AB}}+\frac{1}{\sqrt{AC}}\right)^2.$$Prove that $AE + AF = BC$
A convex $n$-gon encloses a circle of radius $ r$, and is itself enclosed within a circle of radius $R$. Prove that
$$\frac{r}{R} \le \cos \frac{180^o}{n}.$$
In an acute triangle $ABC$, the segment $CD$ is an altitude and $H$ is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle $DHB$, determine all possible values of $\angle CAB$.
Given an acute-angled triangle, describe all interior points whose orthogonal projections on the sides form a triangle similar to the original one.
The points $M$ and $N$ are chosen on the angle bisector $AL$ of a triangle $ABC$ such that $\angle ABM=\angle ACN=23^{\circ}$. $X$ is a point inside the triangle such that $BX=CX$ and $\angle BXC=2\angle BML$. Find $\angle MXN$.
Let $ABC$ be a scalene and non-right triangle. Let $A'$ be the second intersection point of the median drawn from $A$ with the circumcircle of the triangle. Let the tangents to the circumcircle of $ABC$ at points $A$ and $A'$ intersect at $A"$ . Similarly define points $B"$ and $C"$. Prove that $A"$, $B "$, $C"$ are collinear.
Let $ABC$ be a given triangle. Let $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ be circles with radius $p$, centers $A', B',C'$ respectively, and both the legs of angle $\angle BAC$ are tangents to $\Gamma_A$, both legs of $\angle ABC$ are tangents to $\Gamma_B$, both legs of angle $\angle ABC$ are tangents to$\Gamma_C$. The circle $\Gamma$ touches each of the circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ in exactly one point such that all three circles are inside of $\Gamma$, or they are all outside of $\Gamma$. Let $O', I$ and $O$ be the center of $\Gamma$, the incenter of triangle $ABC$ and the circumcenter of triangle $ABC$, respectively. Show that $O'$ lies on the line $IO$.
Let $M$ be the centroid of a non-equilateral triangle $ABC$. Let $A$ and $A'$ lie on opposite sides of the line $BC$ such that the triangle $BCA'$ is equilateral, and let $A''$ be such an internal point of the segment $AA'$ that $A''A' =2AA''$. Let the points $B',B'',C',C''$ be defined analogously. Prove that the triangle $A''B''C''$ is equilateral centered at $M$ .
The point $L$ is the internal point of the side $AC$ of the isosceles triangle $ABC$ ($AB = BC$). The circle $\omega$ goes through $B$ and is tangent to $AC$ at $L$. It intersects the line $AB$ at points $B$ and $D$ and the line $BC$ at points $B$ and $E$. Let $M$ be the midpoint of the segment $DE$ and let $N\ne L$ be the intersection of the lines $BM$ and $AC$. Given that $\frac{AN}{CN}=\frac{AL}{CL}> 1$ prove that the angle $A LB$ equals $60^o$.
problem 13 Let $E$ be an interior point of the convex quadrilateral $ABCD$. Construct triangles $\triangle ABF,\triangle BCG,\triangle CDH$ and $\triangle DAI$ on the outside of the quadrilateral such that the similarities $\triangle ABF\sim\triangle DCE,\triangle BCG\sim \triangle ADE,\triangle CDH\sim\triangle BAE$ and $ \triangle DAI\sim\triangle CBE$ hold. Let $P,Q,R$ and $S$ be the projections of $E$ on the lines $AB,BC,CD$ and $DA$, respectively. Prove that if the quadrilateral $PQRS$ is cyclic, then
\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]
Let $P$ be a point inside a square $ABCD$ such that $PA:PB:PC$ is $1:2:3$. Determine the angle $\angle BPA$.
Let $ABC$ be an acute triangle and $D$ an interior point of its side AC. We call a side of the triangle $ABD$ friendly if the excircle of $ABD$ tangent to that side has its centre on the circumcircle of $ABC$. Prove that there are exactly two friendly sides of $ABD$ if and only if $BD = CD$.
Let $AB$ and $CD$ be two diameters of the circle $C$. For an arbitrary point $P$ on $C$, let $R$ and $S$ be the feet of the perpendiculars from $P$ to $AB$ and $CD$, respectively. Show that the length of $RS$ is independent of the choice of $P$.
Let $A$ and $B$ be two circles, external to each other. Let $\ell$ be a line not meeting the circles. For any point $X$ on $\ell$, let $E$ be a point of contact of a tangent to $A$ through $X$ , and $F$ a point of contact of a tangent to $B$ through $X$ . Find the position of $X$ on $\ell$ is minimized such that $E X + F X$ is minimized.
A circulator is an instrument which draws the circumcircle of three given points in the plane (if the points happen to be collinear, it draws the line through them). Is it possible to construct, only with the help of a circulator, the centre of a given circle?
Let $a, b$, and $c$ be the lengths of the sides of a triangle, $R$ the radius of its circumcircle, and $ r$ the radius of its incircle. Prove that$$\frac{Rr}{(a + b + c)^2} \le \frac{1}{54}$$
The incircle of a triangle $ABC$ touches the sides $BC,CA,AB$ at $D,E,F$, respectively. Let $G$ be a point on the incircle such that $FG$ is a diameter. The lines $EG$ and $FD$ intersect at $H$. Prove that $CH\parallel AB$.
Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality
\[AE\cdot ED + BE^2=CD\cdot AE.\]
Show that $\angle EBA=\angle DCB$.
Let $\Gamma$ be a circle, and $A$ a point outside $\Gamma$. For a point $B$ on $\Gamma$, let $C$ be the third vertex of the equilateral triangle $ABC$ (with vertices $A, B$ and $C$ going clockwise). Find the path traced out by $C$ as $B$ moves around $\Gamma$.
Suppose that the quadrilateral $ABCD$ satisfies $\angle ABD=30^o$, $\angle CDB = 20^o$ and $\angle BCA = \angle ACD = 40^o$. Determine $\angle DAC$.
The side of a triangle is subdivided by the bisector of its opposite angle into two segments of lengths $1$ and $3$. Determine all possible values of the area of that triangle.
Two disks are placed inside a square. What is the maximal proportion of the square that can be covered by the disks, if they are not permitted to overlap? Is it possible to cover more if overlap is allowed?
Consider a right angled triangle $ABC$ with sides of length $3, 4$, and $5$. Determine the greatest possible radius of a circle that is tangent to two among the lines $BC$, $CA$, and $AB$ and that in addition passes through at least one of the points $A, B$, and $C$.
In an acute triangle $ABC$ with $AC > AB$, let $D$ be the projection of $A$ on $BC$, and let $E$ and $F$ be the projections of $D$ on $AB$ and $AC$, respectively. Let $G$ be the intersection point of the lines $AD$ and $EF$. Let $H$ be the second intersection point of the line $AD$ and the circumcircle of triangle $ABC$. Prove that\[AG \cdot AH=AD^2\]
Three line segments, all of length $ 1$, form a connected figure on the plane. Any point that is common to two of these line segments is an endpoint of both segments. Find the maximum area of the convex hull of the figure.
A triangle $ABC$ satisfies $AB < AC$. Let $I$ be the center of the excircle tangent to the side $AC$. Point $P$ lies inside of the angle $BAC$, but outside of the triangle $ABC$ and satisfies $ \angle CPB = \angle PBA + \angle ACP$. Prove that $AP \le AI$.
A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.
$D$ is a point inside triangle $ABC$. The circle $S_1$ inscribed in the triangle $ABD$ touches the circle $S_2$ inscribed in the triangle $CBD$. Prove that the intersection point of outer common tangent lines of circles $S_1$ and $S_2$ lies on the line $AC$.
$A$ and $B$ are points on a given circle. Points $C$ and $D$ move along the circle such that $C$ and $D$ are on the same side of the line $AB$ and the length of the segment $CD$ does not change. $I_1$ and $I_2$ are incenters of the triangles $ABC$ and $ABD$. Prove that there exists a circle such that in every moment the line $I_1I_2$ touches this circle.
Circles $S_1$ and $S_2$ intersect in points $P$ and $Q$ and lay inside an inscribed quadrilateral $ABCD$. $S_1$ touches the sides $AB$, $BC$ and $AD$.$S_2$ touches the sides $CD, BC$ and $AD$. The lines $PQ, AB, CD$ meet in one point. Prove that $BC \parallel AD$.
Consider a triangle $ABC$, satisfying $BC < \frac{AC+AB}{2}$. Prove that$ \angle BAC < \frac{\angle CBA+\angle ACB}{2}$.
All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron.
Let $ABC$ be a triangle, and let $X, Y , Z$ be points on $BC$, $CA$, $AB$, respectively. Suppose that $AX$, $BY$ and $CZ$ intersect in a point $P$. Prove that$$\frac{AP}{AX}+\frac{BP}{BY}+\frac{CP}{CZ}= 2.$$
Circles $\alpha$ and $\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\alpha$ and $\beta$. A third circle of the same radius passes through $P$ and intersects $\alpha$ and $\beta$ in the points $X$ and $Y$ , respectively. Show that the line $XY$ is parallel to the line $AB$.
A circle $\omega$ is tangent to the side $BC$ of a triangle $ABC$ at point $T$. The side $AB$ intersects $\omega$ at points $P$ and $R$ ($A$ is closer to $P$ than $R$); the side $AC$ intersects $\omega$ at points $Q$ and $S$ ($A$ is closer to $Q$ than $S$). The lines $AT$, $BQ$ and $CP$ are concurrent. Prove that the lines $AT$, $BS$ and $CR$ are also concurrent.
$A$ and $B$ are two convex polygons without common points. None of them is fully contained inside the other one. Prove that there exists such a line $\ell$ that does not intersect each of the polygons and $A$ and $B$ lie on the different sides of $\ell$.
Four circles in a plane have a common center. Their radii form a strictly increasing arithmetic progression. Prove that there is no square with each vertex lying on a different circle.
Let $\Gamma$ be the circumcircle of an acute triangle $ABC.$ The perpendicular to $AB$ from $C$ meets $AB$ at $D$ and $\Gamma$ again at $E.$ The bisector of angle $C$ meets $AB$ at $F$ and $\Gamma$ again at $G.$ The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I.$ Prove that $AI = EB.$
Let $ABC$ be a triangle with circumcircle $\omega$. Let $D, E$ and $F$ be points on the sides $BC$, $CA$ and $AB$ such that the circumcircle of the triangle $DEF$ touches $\omega$ at $A$. Let $G$ and $H$ be the intersection points of the circumcircles of the triangles $BDE$ and $CD F$ with $\omega$ (different from $B, C$), respectively. Prove that the lines $GE$ and $HF$ intersect on $AD$.
Triangle $ABC$ is given. Let $M$ be the midpoint of the segment $AB$ and $T$ be the midpoint of the arc $BC$ not containing $A$ of the circumcircle of $ABC.$ The point $K$ inside the triangle $ABC$ is such that $MATK$ is an isosceles trapezoid with $AT\parallel MK.$ Show that $AK = KC.$
Points $X$ , $Y$, $Z$ lie on a line $k$ in this order. Let $\omega_1$, $\omega_2$, $\omega_3$ be three circles of diameters $XZ$, $XY$ , $YZ$ , respectively. Line $\ell$ passing through point $Y$ intersects $\omega_1$ at points $A$ and $D$, $\omega_2$ at $B$ and $\omega_3$ at $C$ in such manner that points $A, B, Y, X, D$ lie on $\ell$ in this order. Prove that $AB =CD$.
Let $ABCD$ be a square inscribed in a circle $\omega$ and let $P$ be a point on the shorter arc $AB$ of $\omega$. Let $CP\cap BD = R$ and $DP \cap AC = S.$
Show that triangles $ARB$ and $DSR$ have equal areas.
Let $ABC$ be a triangle with $A B \ne AC$. The angle bisector of $\angle BAC$ intersects $BC$ in $D$. The circle with diameter $AD$ intersects $AC$ again in $F$, and $BC$ again in $Q$. The point $R\ne Q$ lies on the line parallel to $AD$ through $Q$. Suppose that $AQ = AR$. Prove that the points $B, F, Q$, and $R$ lie on a common circle.
Let $ABCD$ be a convex quadrilateral such that the line $BD$ bisects the angle $ABC.$ The circumcircle of triangle $ABC$ intersects the sides $AD$ and $CD$ in the points $P$ and $Q,$ respectively. The line through $D$ and parallel to $AC$ intersects the lines $BC$ and $BA$ at the points $R$ and $S,$ respectively. Prove that the points $P, Q, R$ and $S$ lie on a common circle.
Let $AB$ be a common chord of different circles $\Gamma_1$ and $\Gamma_2$ and let $P$ be a point not collinear with $A$ and $B$. Assume that the line $AP$ meets $\Gamma_1$ and $\Gamma_2$ again in $K$ and $L$, respectively, the line $BP$ meets $\Gamma_1$ and $\Gamma_2$ again in $M$ and $N$ , respectively, and that all the points mentioned so far are different. Let $O_1$ and $O_2$ be the circumcenters of triangles $KMP$ and $LNP$ , respectively. Prove that $O_1O_2$ is perpendicular to $AB$.
Diagonal $AC$ of a convex quadrilateral $ABCD$ is a bisector of angle $\angle A$ and $\angle A+\angle C = 90^o$. A point $P$ on the segment $AC$ is inside the triangle $ABD$ and is such that $\angle BPD = 90^o$. $CQ$ is a diameter of the circumcircle $CBD$. Prove that the line $PQ$ passes through the midpoint of arc $DAB$.
The quadrilateral $Q$ has a longest side of length $ b$ and a shortest side of length $a$. Form a new quadrilateral $Q'$ by joining the successive midpoints of the edges of $Q$. Supposing that $Q$ and $Q'$ are similar, prove that $\frac{b}{a}< 1 +\sqrt2$.
The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^{\circ} .$ Prove that\[AB \cdot CD + AD \cdot BC < AC(AB + AD).\]
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$.
Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)
A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\angle LKN = \angle ACB$. Find $\angle BAC $ given that the triangle $CKN$ is equilateral.
Let $CM$ be a median of $\vartriangle ABC$. The circle with diameter $CM$ intersects segments $AC$ and $BC$ in points $P$ and $Q$, respectively. Given that $AB\parallel PQ$ and $\angle BAC = \alpha$, what are the possible values of $\angle ACB$?
For what positive numbers$ m$ and $n$ do there exist points $A_1, ..., Am$ and $B_1 ..., B_n$ in the plane such that, for any point $P$, the equation$$|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2 $$holds true?
Let $D$ be the footpoint of the altitude from $B$ in the triangle $ABC$ , where $AB=1$ . The incircle of triangle $BCD$ coincides with the centroid of triangle $ABC$. Find the lengths of $AC$ and $BC$.
Suppose that $A, B, C$, and $X$ are any four distinct points in the plane with$$\max \,(BX,CX) \le AX \le BC.$$Prove that $\angle BAC \le 150^o$.
Let $ABC$ be a scalene triangle. Let $D$ and $E$ be the points where the incircle touches sides $BC$ and $CA$, respectively. Let $K$ be the common point of line $BC$ and the bisector of the angle $\angle BAC$. Let $AD$ intersect $EK$ in $P$. Prove that $PC$ is perpendicular to $AK$.
Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$. Let $P$ be a variable point on that arc $BC$ not containing the points $A$ and $D$. Suppose $BC$ intersects the lines $AP$ and $DP$ in $X$ and $Y$, respectively. Show that, if we choose $P$ in such a way as to maximise the length of the segment $XY$, then $BX = CY$.
In the non-isosceles triangle $ABC$ an altitude from $A$ meets side $BC$ in $D$ . Let $M$ be the midpoint of $BC$ and let $N$ be the reflection of $M$ in $D$ . The circumcirle of triangle $AMN$ intersects the side $AB$ in $P\ne A$ and the side $AC$ in $Q\ne A$ . Prove that $AN,BQ$ and $CP$ are concurrent.
In triangle $ABC$, the interior and exterior angle bisectors of $ \angle BAC$ intersect the line $BC$ in $D $ and $E$, respectively. Let $F$ be the second point of intersection of the line $AD$ with the circumcircle of the triangle $ ABC$. Let $O$ be the circumcentre of the triangle $ ABC $and let $D'$ be the reflection of $D$ in $O$. Prove that $ \angle D'FE =90.$
Let $\vartriangle ABC$ be a triangle, and let $P$ and $Q$ be two distinct points on the tangent line at $A$ to the circumscribed circle. They are such that $|AP| = |AQ|$ and that $BP$ and $CQ$ meet inside the triangle. Let $S$ be a point inside triangle $\vartriangle ABC$ such that $\angle ABP = \angle BCS$ and $\angle ACQ = \angle CBS$. Prove that $AS$ is the median of $\vartriangle ABC$ through $A$.
Let $ABC$ be a triangle. Let its altitudes $AD$, $BE$ and $CF$ concur at $H$. Let $K, L$ and $M$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Prove that, if $\angle BAC = 60^o$, then the midpoints of the segments $AH$, $DK$, $EL$, $FM$ are concyclic.
In triangle $ABC,$ the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $AB$ and $AC,$ respectively. Points $F$ and $G$ on the extensions of $AB$ and $AC$ beyond $B$ and $C,$ respectively, satisfy $BF = CG = BC.$ Prove that $F G \parallel DE.$
Let $ABCD$ be a convex quadrilateral with $AB = AD.$ Let $T$ be a point on the diagonal $AC$ such that $\angle ABT + \angle ADT = \angle BCD.$ Prove that $AT + AC \geq AB + AD.$
Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$
Let $ABC$ be a triangle and $D$ and $E$ points on the lines $CA$ and $BA$ such that $CD = AB$ , $BE = AC$ and $A$, $D$ and $E$ lie on the same side of $BC$ . Let $I$ be the incenter of $ABC$ and let $H$ the orthocenter of $BCI$. Show that $D$, $E$ and $H$ are collinear.
Consider triangles in the plane where each vertex has integer coordinates. Such a triangle can be legally transformed by moving one vertex parallel to the opposite side to a different point with integer coordinates. Show that if two triangles have the same area, then there exists a series of legal transformations that transforms one to the other.
Let $ABCD$ be a convex quadrilateral. Let $P$ be a point such that $\angle APC = \angle BPD=30^o$. Prove that$$2(AB + AC + AD + BC + BD + CD) \ge PA + PB + PC + PD.$$
Let $ABCD$ be a cyclic quadrilateral with $AB$ and $CD$ not parallel. Let $M$ be the midpoint of $CD.$ Let $P$ be a point inside $ABCD$ such that $P A = P B = CM.$ Prove that $AB, CD$ and the perpendicular bisector of $MP$ are concurrent.
Let $ABC$ be a triangle and let $P$ be a point such that $AP$ is the angle bisector of $\angle BAC$ and segment $BC$ bisects segment $AP$. Prove that perimeter of triangle $ABC$ is greater than or equal to perimeter of triangle $PBC$,
Starting with three points $A,B,C$ in general position and the circumcircle $k$ of $\vartriangle ABC$, a step consists of drawing a line $\ell$ and obtaining all points of intersection of $\ell$ with the lines already drawn and with $k$, where $\ell$ is
(1) the line through two distinct points that had been obtained before or
(2) the bisector of an angle $\angle XYZ$ , where $X,,Z$ are three previously obtained distinct points on $k$.
Is it always (i.e., for any choice of $A,B,C$) possible to obtain the orthocenter of $\vartriangle ABC$ in a finite number of steps?
Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$ , respectively, of an acute triangle $ABC$ . Let $\omega_B$ be the circle centered at $M$ passing through $B$, and let $\omega_C$ be the circle centered at $N$ passing through $C$ . Let the point $D$ be such that $ABCD$ is an isosceles trapezoid with $AD$ parallel to $BC$ . Assume that $\omega_B$ and $\omega_C$ intersect in two distinct points $P$ and $Q$. Show that $D$ lies on $PQ$.
Let $D_a$, $E_b$, and $F_c$ be three collinear points on the sides $BC$, $CA$, and $AB$ of triangle $ABC$. Let $D_b$ and $D_c$ be on $BC$ such that $D_bE_b$ and $D_cF_c$ are parallel. Let $E_c$ and $E_a$ be on $CA$ such that $E_cF_c$ and $E_aD_a$ are parallel. Let $F_a$ and $F_b$ be on $AB$ such that $F_aD_a$ and $F_bE_b$ are parallel. Show that the three midpoints of $E_aF_a$ , $F_bD_b$, and $D_cE_c$ are collinear.
Let $H$ and $I$ be the orthocenter and incenter, respectively, of an acute-angled triangle $ABC$. The circumcircle of the triangle $BCI$ intersects the segment $AB$ at the point $P$ different from $B$. Let $K$ be the projection of $H$ onto $AI$ and $Q$ the reflection of $P$ in $K$. Show that $B$, $H$ and $Q$ are collinear.
Let $ABC$ be an acute triangle with circumcircle $\Gamma$, circumcenter $O$, and $AB < AC$. Let $D$ be the midpoint of $AB$, and $E$ a point on $AC$ such that $BE = CE$. The circumcircle $\omega$ of $BDE$ intersects $\Gamma$ at $F\ne B$. Let $K$ be the orthogonal projection of $B$ onto $AO$, and suppose that $A$ and $K$ lie on different sides of $BE$. Show that the lines $DF$ and $CK$ intersect at a point on $\Gamma$.
Triangle $ABC$ has angles $\angle A = 80^o$, $\angle B = 70^o$ and $\angle C = 30^o$. Point $P$ lies on the angle bisector at $A$, satisfying $\angle BPC = 130^o$. Drop perpendiculars $PX, PY, PZ$ onto the edges $BC, AC, AB$, respectively. Prove that$$AY^3 + BZ^3 + CX^3 = AZ^3 + BX^3 + CY^3.$$
Line \(\ell\) touches circle $S_1$ in the point $X$ and circle $S_2$ in the point $Y$. We draw a line $m$ which is parallel to $\ell$ and intersects $S_1$ in a point $P$ and $S_2$ in a point $Q$. Prove that the ratio $XP/YQ$ does not depend on the choice of $m$.
$AD$, $BE$ and $CF$ are heights of triangle $ABC$ with obtuse angle $B$. Point $T$ is midpoint of the segment $AD$, point $S$ is midpoint of the segment $CF$. Point $M$ is symmetrical to $T$ with respect to line $BE$. Point $N$ is symmetrical to $T$ with respect to line $BD$. Prove that the circumcircle of triangle $BMN$ passes through $S$.
Let $ABC$ be an isosceles triangle with $AB = AC$. Let P be a point in the interior of $ABC$ such that $PB > PC$ and $\angle PBA = \angle PCB$. Let $M$ be the midpoint of the side $BC$. Let $O$ be the circumcenter of the triangle $APM$. Prove that $\angle OAC=2 \angle BPM$ .
Let $ABC$ be a triangle in which $\angle ABC = 60^{\circ}$. Let $I$ and $O$ be the incentre and circumcentre of $ABC$, respectively. Let $M$ be the midpoint of the arc $BC$ of the circumcircle of $ABC$, which does not contain the point $A$. Determine $\angle BAC$ given that $MB = OI$.
Let $ABCD$ be a parallelogram. Points $X$ and $Y$ lie on the sides $BC$ and $CD$, respectively. Segments $BY$ and $DX$ intersect each other at $P$. Prove that the line passing through midpoints of the segments $BD$ and $XY$ is parallel to $AP$.
(version 1) Let $ABC$ be a triangle with incenter $I$ and denote the midpoints on the sides $BC$, $CA$ and $AB$ by $M_A$ ,$M_B$ and $M_C$ respectively. The point $K$ is the reflection of $I$ over $M_A$ , and $\ell$ is the line through $K$ parallel with $BC$. We define $P$ and $Q$ as the intersections of $\ell$ with $M_AM_B$ and $M_AM_C$ respectively. Show that the excircle opposite $M_A$ of $\vartriangle PM_AQ$ goes through the $A$-excenter of $\vartriangle ABC$ .
(version 2) Let $ABCD$ be a parallelogram where $M$ is the midpoint of $BD$ . Let $I$ be the incenter of $\vartriangle ABD $, and $\ell$ the line through $I$ parallel with $BD$. We let $M_D$ , $M_B$ be the midpoints of $AB$, $AD$. Show that if $P = MM_D \cap \ell$ and $Q = MM_B \cap \ell$ then the excircle of $\vartriangle PMQ$ opposite $M$ passes through the $C$-excenter of $\vartriangle BCD$ .
Let $I_A$ $,I_B$ and $I_c$ be the excenters opposite of $A, B$ and $C$, respectively, of $\vartriangle ABC$ . Prove that the Euler lines of triangles $BI_A C$ ,$C I_B A$ and $A I_C B$ all concur on the circumcircle of $\vartriangle A B C$ .
Let $P$ be a point inside the acute angle $\angle BAC$. Suppose that $\angle ABP = \angle ACP = 90^{\circ}$. The points $D$ and $E$ are on the segments $BA$ and $CA$, respectively, such that $BD = BP$ and $CP = CE$. The points $F$ and $G$ are on the segments $AC$ and $AB$, respectively, such that $DF$ is perpendicular to $AB$ and $EG$ is perpendicular to $AC$. Show that $PF = PG$.
Let $A'$ be the result of reflection of vertex $A$ of triangle $ABC$ through line $BC$ and let $B'$ be the result of reflection of vertex $B$ through line $AC$. Given that $\angle BA'C = \angle BB'C$ , can the largest angle of triangle $ABC$ be located:
i) At vertex $A$?
ii) At vertex $B$?
iii) At vertex $C$?
Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?
Let $\omega_1$ and $\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively, with $O_2$ lying on $\omega_1$. Let $A$ be a common point of $\omega_1$ and $\omega_2$. A line through $A$ intersects $\omega_1$ in $B \ne A$ and $\omega_2$ in $C\ne A$ such that $A$ lies between $B$ and $C$. The ray $O_2O_1$ intersects $\omega_2$ in $D$ and contains a point $E$ such that $\angle EAD = \angle DCO_2$ and $D$ lies between $O_2$ and $E$. Show that $BO_2$ bisects $CE$.
Let $ABCD$ be a convex quadrilateral such that $|AD| < |AB|$ and $|CD| < |CB|$. Prove that $\angle ABC < \angle ADC$.
Two points $M$ and $N$ have been selected from the edge $BC$ of a tetrahedron $ABCD$ so that the point $M$ lies between $B$ and $N$, the point $N$ lies between $M$ and $C$, and the angle between the planes $AND$ and $ACD$ is equal to the angle between the planes $AMD$ and $ABD$. Prove that$$\frac{MB}{MC}+ \frac{NB}{NC} \ge 2\frac {|\vartriangle ABD|}{|\vartriangle ACD|}$$where$ |\vartriangle ABD |$ is the area of the triangle $\vartriangle ABD$ , and similarly $|\vartriangle ACD|$ is the area of $\vartriangle ACD$ .
Let $ABC$ be a triangle with $AB = AC$. Let $M$ be the midpoint of $BC$. Let the circles with diameters $AC$ and $BM$ intersect at points $M$ and $P$. Let $MP$ intersect $AB$ at $Q$. Let $R$ be a point on $AP$ such that $QR \parallel BP$. Prove that $CP$ bisects $\angle RCB$.
Let $ABC$ be a scalene triangle. Let $P$ be an interior point of $ABC$ such that $AP \perp BC$. Assume that $BP$ and $CP$ intersect $AC$ and $AB$ at $X$ and $Y$, respectively. Prove that $AX = AY$ iff there exists a circle with centre lying on $BC$ and tangent to $AB$ and $AC$ at points $Y$ and $X$, respectively.
Let $ABC$ be a triangle and $H$ its orthocenter. Let $D$ be a point lying on the segment $AC$ and let $E$ be the point on the line $BC$ such that $BC\perp DE$. Prove that $EH\perp BD$ if and only if $BD$ bisects $AE$.
Let $ABC$ be a triangle and $H$ its orthocenter. Let $P$ be an arbitrary point on segment $BC$. Prove that $H$ lies on the line passing through reflections of $P$ with respect to $AB$ and $AC$ iff triangle $ABC$ is right angled.
Let $ABCDEF$ be a convex hexagon in which $AB=AF$, $BC=CD$, $DE=EF$ and $\angle ABC = \angle EFA = 90^{\circ}$. Prove that $AD\perp CE$.
Let $X, Y$ be points on $AB, AC$ of triangle $ABC$, respectively such that $B, C, X, Y$ lie on one circle. The median of triangle $ABC$ from $A$ intersects perpendicular bisector of $XY$ at $P$. Find $\angle BAC$, if $PXY$ is equilateral.
A circle with center $O$ passes through the vertices $B$ and $C$ of the triangle $ABC$ and intersects for the second time segments $AB$ and $AC$ in points $C_1$ and $B_1$ correspondingly. The lines $AO$, $BB_1$ and $CC_1$ are concurrent. Prove that triangle $ABC$ is isosceles.
Let $ABC$ be a triangle with $\angle ABC = 90^{\circ}$, and let $H$ be the foot of the altitude from $B$. The points $M$ and $N$ are the midpoints of the segments $AH$ and $CH$, respectively. Let $P$ and $Q$ be the second points of intersection of the circumcircle of the triangle $ABC$ with the lines $BM$ and $BN$, respectively. The segments $AQ$ and $CP$ intersect at the point $R$. Prove that the line $BR$ passes through the midpoint of the segment $MN$.
Let $n \geq 4$, and consider a (not necessarily convex) polygon $P_1P_2\hdots P_n$ in the plane. Suppose that, for each $P_k$, there is a unique vertex $Q_k\ne P_k$ among $P_1,\hdots, P_n$ that lies closest to it. The polygon is then said to be hostile if $Q_k\ne P_{k\pm 1}$ for all $k$ (where $P_0 = P_n$, $P_{n+1} = P_1$).
(a) Prove that no hostile polygon is convex.
(b) Find all $n \geq 4$ for which there exists a hostile $n$-gon.
Let $ABC$ be a triangle with $AB > AC$. The internal angle bisector of $\angle BAC$ intersects the side $BC$ at $D$. The circles with diameters $BD$ and $CD$ intersect the circumcircle of $\triangle ABC$ a second time at $P \not= B$ and $Q \not= C$, respectively. The lines $PQ$ and $BC$ intersect at $X$. Prove that $AX$ is tangent to the circumcircle of $\triangle ABC$.
Let $ABCDE$ be a convex pentagon inscribed in a circle $\omega$ such that $CD \parallel BE$. The line tangent to $\omega$ at $B$ intersects the line $AC$ at a point $F$ such that $A$ lies between the points $C$ and $F$. The lines $BD$ and $AE$ intersect at $G$. Prove that the line $FG$ is tangent to the circumcircle of $ADG$.
Let $ABC$ be a triangle with circumcircle $\omega$. The internal angle bisectors of $\angle ABC$ and $\angle ACB$ intersect $\omega$ at $X\neq B$ and $Y\neq C$, respectively. Let $K$ be a point on $CX$ such that $\angle KAC = 90^\circ$. Similarly, let $L$ be a point on $BY$ such that $\angle LAB = 90^\circ$. Let $S$ be the midpoint of arc $CAB$ of $\omega$. Prove that $SK=SL$.
Let $\omega$ be a semicircle with diameter $XY$ . Let $M$ be the midpoint of $XY$ . Let $A$ be an arbitrary point on $\omega$ such that $AX < AY$ . Let $B$ and $C$ be points lying on the segments $XM$ and $YM$, respectively, such that $BM = CM$. The line through $C$ parallel to $AB$ intersects $\omega$ at $P$. The line through $B$ parallel to $AC$ intersects $\omega$ at $Q$. The line $PQ$ intersects the line $XY$ at a point $S$. Prove that the line $AS$ is tangent to $\omega$
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $AC$, respectively. Let $H$ be the orthocenter of $BXY$. Let $CH$ intersect $\ell$ at $D$. Prove that $BA$ bisects angle $CBD$.
Let $\omega$ and $I$ be the incircle and the incentre of a triangle $ABC$, respectively. Let $E$ and $F$ be the tangency points of $\omega$ with the sides $AC$ and $AB$, respectively. The perpendicular bisector of $AI$ intersects $AC$ at a point $P$. Point $Q$ lies on $AB$ and satisfies $QI \perp FP$. Prove that $EQ \perp AB$.
An acute triangle $ABC$ is given and let $H$ be its orthocenter. Let $\omega$ be the circle through $B$, $C$ and $H$, and let $\Gamma$ be the circle with diameter $AH$. Let $X\neq H$ be the other intersection point of $\omega$ and $\Gamma$, and let $\gamma$ be the reflection of $\Gamma$ over $AX$.
Suppose $\gamma$ and $\omega$ intersect again at $Y\neq X$, and line $AH$ and $\omega$ intersect again at $Z \neq H$. Show that the circle through $A,Y,Z$ passes through the midpoint of segment $BC$.
Consider the Euclidean plane, the points $A = (0, 0)$ and $B = (1, 0)$ and the open half-strip$$S =\{(x, y) : 0 < x < 1, y > 0\}$$with width $ 1$ and vertices $A$ and $B$.
Find all functions $f : S \to S$ satisfying the following conditions for all $P,Q \in S$:
(i) $f(f(P)) = P$
(ii) If $P, Q,A$ are collinear, then $f(P), f(Q)$ and $B$ are collinear.
(iii) If$ f(P) = P$ and $f(Q) = Q$, then there is a circle containing $A, B, P$ and $Q$.
Let $A, B, C, P$ and $Q$ five pairwise different points in the plane. Suppose that $A, B$ and $C$ are not collinear and that
$$\frac{AP}{BP}= \frac{AQ}{BQ}= \frac{21}{20} \,\, , \frac{BP}{CP}= \frac{BQ}{CQ}= \frac{20}{19}.$$Prove that the line $PQ$ contains the circumcentre of the triangle $ABC$.
On a plane, Bob chooses 3 points $A_0$, $B_0$, $C_0$ (not necessarily distinct) such that $A_0B_0+B_0C_0+C_0A_0=1$. Then he chooses points $A_1$, $B_1$, $C_1$ (not necessarily distinct) in such a way that $A_1B_1=A_0B_0$ and $B_1C_1=B_0C_0$.
Next he chooses points $A_2$, $B_2$, $C_2$ as a permutation of points $A_1$, $B_1$, $C_1$. Finally, Bob chooses points $A_3$, $B_3$, $C_3$ (not necessarily distinct) in such a way that $A_3B_3=A_2B_2$ and $B_3C_3=B_2C_2$. What are the smallest and the greatest possible values of $A_3B_3+B_3C_3+C_3A_3$ Bob can obtain?
Figure shows two non-intersecting circles $\alpha$ and $\beta$ in space. We say that circle $\alpha$ devours circle $\beta$ since one chord of $\beta$ (solid) is strictly contained in a chord of $\alpha$ (dashed).
The question is whether it is possible to place three circles $\alpha$, $\beta$ and $\gamma$ in space so as to have $\alpha$ devouring $\beta$ devouring $\gamma$ devouring $\alpha$. The radii of the circles need not be equal.
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