Found problems: 41
2020 Turkey Junior National Olympiad, 3
The circumcenter of an acute-triangle $ABC$ with $|AB|<|BC|$ is $O$, $D$ and $E$ are midpoints of $|AB|$ and $|AC|$, respectively. $OE$ intersects $BC$ at $K$, the circumcircle of $OKB$ intersects $OD$ second time at $L$. $F$ is the foot of altitude from $A$ to line $KL$. Show that the point $F$ lies on the line $DE$
2011 Indonesia TST, 3
Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and
intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at
points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and
touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$
and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle
$\omega$ are also collinear.
1982 IMO Shortlist, 5
The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.
1996 Mexico National Olympiad, 6
In a triangle $ABC$ with $AB < BC < AC$, points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp CA$ and $BB'=CA$, and $CC' \perp AB$ and $CC'= AB$, as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C' $ are collinear.
1986 IMO Shortlist, 14
The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.
2025 Israel National Olympiad (Gillis), P2
Let $ABCD$ be a rhombus. Eight additional points $X_1$, $X_2$, $Y_1$, $Y_2$, $Z_1$, $Z_2$, $W_1$, $W_2$ were chosen so that the quadrilaterals $AX_1BX_2$, $BY_1CY_2$, $CZ_1DZ_2$, $DW_1AW_2$ are squares. Prove that the eight new points lie on two straight lines.
2023 Sharygin Geometry Olympiad, 23
An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.
2022 Romania Team Selection Test, 3
Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$
and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with
center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the
midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second
time at $Y$, show that $A, Y$, and $M$ are collinear.
[i]Proposed by Nikola Velov, North Macedonia[/i]
1987 IMO Shortlist, 12
Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear.
[i]Proposed by Poland.[/i]
1986 IMO Longlists, 44
The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.
2016 Harvard-MIT Mathematics Tournament, 10
Let $ABC$ be a triangle with incenter $I$ whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$. Point $P$ lies on $\overline{EF}$ such that $\overline{DP} \perp \overline{EF}$. Ray $BP$ meets $\overline{AC}$ at $Y$ and ray $CP$ meets $\overline{AB}$ at $Z$. Point $Q$ is selected on the circumcircle of $\triangle AYZ$ so that $\overline{AQ} \perp \overline{BC}$.
Prove that $P$, $I$, $Q$ are collinear.
2021 2nd Memorial "Aleksandar Blazhevski-Cane", 5
Let $\triangle ABC$ be a triangle with circumcenter $O$. The perpendicular bisectors of the segments $OA,OB$ and $OC$ intersect the lines $BC,CA$ and $AB$ at $D,E$ and $F$, respectively. Prove that $D,E,F$ are collinear.
1982 IMO Longlists, 37
The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.
2010 Korea Junior Math Olympiad, 3
In an acute triangle $\triangle ABC$, let there be point $D$ on segment $AC, E$ on segment $AB$ such that $\angle ADE = \angle ABC$. Let the bisector of $\angle A$ hit $BC$ at $K$. Let the foot of the perpendicular from $K$ to $DE$ be $P$, and the foot of the perpendicular from $A$ to $DE$ be $L$. Let $Q$ be the midpoint of $AL$. If the incenter of $\triangle ABC$ lies on the circumcircle of $\triangle ADE$, prove that $P,Q$ and the incenter of $\triangle ADE$ are collinear.
Mathematical Minds 2024, P8
Let $ABC$ be a triangle with circumcircle $\Omega$, incircle $\omega$, and $A$-excircle $\omega_A$. Let $X$ and $Y$ be the tangency points of $\omega_A$ with $AB$ and $AC$. Lines $XY$ and $BC$ intersect in $T$. The tangent from $T$ to $\omega$ different from $BC$ intersects $\omega$ at $K$. The radical axis of $\omega_A$ and $\Omega$ intersects $BC$ in $S$. The tangent from $S$ to $\omega_A$ different from $BC$ intersects $\omega_A$ at $L$. Prove that $A$, $K$ and $L$ are collinear.
[i]Proposed by Ana Boiangiu[/i]
2018 Brazil Team Selection Test, 4
In a triangle $ABC$, points $H, L, K$ are chosen on the sides $AB, BC, AC$, respectively, so that $CH \perp AB$, $HL \parallel AC$ and $HK \parallel BC$. In the triangle $BHL$, let $P, Q$ be the feet of the heights from the vertices $B$ and $H$. In the triangle $AKH$, let $R, S$ be the feet of the heights from the vertices $A$ and $H$. Show that the four points $P, Q, R, S$ are collinear.
2018 Peru Iberoamerican Team Selection Test, P9
Let $\Gamma$ be the circumcircle of a triangle $ABC$ with $AB <BC$, and let $M$ be the midpoint from the side $AC$ . The median of side $AC$ cuts $\Gamma$ at points $X$ and $Y$ ($X$ in the arc $ABC$). The circumcircle of the triangle $BMY$ cuts the line $AB$ at $P$ ($P \ne B$) and the line $BC$ in $Q$ ($Q \ne B$). The circumcircles of the triangles $PBC$ and $QBA$ are cut in $R$ ($R \ne B$). Prove that points $X, B$ and $R$ are collinear.
2011 Korea Junior Math Olympiad, 2
Let $ABCD$ be a cyclic quadrilateral inscirbed in circle $O$. Let the tangent to $O$ at $A$ meet $BC$ at $S$, and the tangent to $O$ at $B$ meet $CD$ at $T$. Circle with $S$ as its center and passing $A$ meets $BC$ at $E$, and $AE$ meets $O$ again at $F(\ne A)$. The circle with $T$ as its center and passing $B$ meets $CD$ at $K$. Let $P = BK \cap AC$. Prove that $P,F,D$ are collinear if and only if $AB = AP$.
2017 Hanoi Open Mathematics Competitions, 14
Given trapezoid $ABCD$ with bases $AB \parallel CD$ ($AB < CD$). Let $O$ be the intersection of $AC$ and $BD$. Two straight lines from $D$ and $C$ are perpendicular to $AC$ and $BD$ intersect at $E$ , i.e. $CE \perp BD$ and $DE \perp AC$ . By analogy, $AF \perp BD$ and $BF \perp AC$ . Are three points $E , O, F$ located on the same line?
2003 Junior Balkan Team Selection Tests - Romania, 1
Suppose $ABCD$ and $AEFG$ are rectangles such that the points $B,E,D,G$ are collinear (in this order). Let the lines $BC$ and $GF$ intersect at point $T$ and let the lines $DC$ and $EF$ intersect at point $H$. Prove that points $A, H$ and $T$ are collinear.
2000 Saint Petersburg Mathematical Olympiad, 10.6
One of the excircles of triangle $ABC$ is tangent to the side $AB$ and to the extensions of sides $CA$ and $CB$ at points $C_1$, $B_1$ and $A_1$ respectively. Another excircle is tangent to side $AB$ and to the extensions of sides $BA$ and $BC$ at points $B_2$, $C_2$ and $A_2$ respectively. Line $A_1B_1$ and $A_2B_2$ intersect at point $P$,. lines $A_1C_1$ and $A_2C_2$ intersect at point $Q$. Prove that the points $A$, $P$, $Q$ are collinear
[I]Proposed by S. Berlov[/i]
2011 Indonesia TST, 3
Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and
intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at
points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and
touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$
and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle
$\omega$ are also collinear.
2019 Thailand Mathematical Olympiad, 8
Let $ABC$ be a triangle such that $AB\ne AC$ and $\omega$ be the circumcircle of this triangle.
Let $I$ be the center of the inscribed circle of $ABC$ which touches $BC$ at $D$.
Let the circle with diameter $AI$ meets $\omega$ again at $K$.
If the line $AI$ intersects $\omega$ again at $M$, show that $K, D, M$ are collinear.
2024 Centroamerican and Caribbean Math Olympiad, 3
Let $ABC$ be a triangle, $H$ its orthocenter, and $\Gamma$ its circumcircle. Let $J$ be the point diametrically opposite to $A$ on $\Gamma$. The points $D$, $E$ and $F$ are the feet of the altitudes from $A$, $B$ and $C$, respectively. The line $AD$ intersects $\Gamma$ again at $P$. The circumcircle of $EFP$ intersects $\Gamma$ again at $Q$. Let $K$ be the second point of intersection of $JH$ with $\Gamma$. Prove that $K$, $D$ and $Q$ are collinear.
2011 Sharygin Geometry Olympiad, 6
In triangle $ABC$ $AA_0$ and $BB_0$ are medians, $AA_1$ and $BB_1$ are altitudes. The circumcircles of triangles $CA_0B_0$ and $CA_1B_1$ meet again in point $M_c$. Points $M_a, M_b$ are defined similarly. Prove that points $M_a, M_b, M_c$ are collinear and lines $AM_a, BM_b, CM_c$ are parallel.