Found problems: 254
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.
2012 Singapore Junior Math Olympiad, 1
Let $O$ be the centre of a parallelogram $ABCD$ and $P$ be any point in the plane. Let $M, N$ be the midpoints of $AP, BP$, respectively and $Q$ be the intersection of $MC$ and $ND$. Prove that $O, P$ and $Q$ are collinear.
2017 Iran Team Selection Test, 3
In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$.
Prove that $X,Y,Z$ are collinear.
[i]Proposed by Hooman Fattahi[/i]
2022 Oral Moscow Geometry Olympiad, 3
Extensions of opposite sides of a convex quadrilateral $ABCD$ intersect at points $P$ and $Q$. Points are marked on the sides of $ABCD$ (one per side), which are the vertices of a parallelogram with a side parallel to $PQ$. Prove that the intersection point of the diagonals of this parallelogram lies on one of the diagonals of quadrilateral $ABCD$.
(E. Bakaev)
Kyiv City MO Seniors 2003+ geometry, 2020.10.5
Given an acute isosceles triangle $ABC, AK$ and $CN$ are its angle bisectors, $I$ is their intersection point . Let point $X$ be the other intersection point of the circles circumscribed around $\vartriangle ABC$ and $\vartriangle KBN$. Let $M$ be the midpoint of $AC$. Prove that the Euler line of $\vartriangle ABC$ is perpendicular to the line $BI$ if and only if the points $X, I$ and $M$ lie on the same line.
(Kivva Bogdan)
2018 Chile National Olympiad, 6
Consider an acute triangle $ABC$ and its altitudes from $A$ ,$B$ that intersect the respective sides at $D ,E$. Let us call the point of intersection of the altitudes $H$. Construct the circle with center $H$ and radius $HE$. From $C$ draw a tangent line to the circle at point $P$. With center $B$ and radius $BE$ draw another circle and from $C$ another tangent line is drawn to this circle in the point $Q$. Prove that the points $D, P$, and $Q$ are collinear.
I Soros Olympiad 1994-95 (Rus + Ukr), 9.7
Given an acute triangle $ABC$, in which $\angle BAC <30^o$. On sides $AC$ and $AB$ are taken respectively points $D$ and $E$ such that $\angle BDC=\angle BDE = \angle ADE = 60^o$. Prove that the centers of the circles. inscribed in triangles $ADE$, $BDE$ and $BCD$ do not lie on the same line.
1967 Spain Mathematical Olympiad, 2
Determine the poles of the inversions that transform four collienar points $A,B, C, D$, aligned in this order, at four points $A' $, $B' $, $C'$ , $D'$ that are vertices of a rectangle, and such that $A'$ and $C'$ are opposite vertices.
2023 Austrian MO Beginners' Competition, 2
Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line.
[i](Walther Janous)[/i]
2010 Junior Balkan Team Selection Tests - Moldova, 3
The tangent to the circle circumscribed to the triangle $ABC$, taken through the vertex $A$, intersects the line $BC$ at the point $P$, and the tangents to the same circle, taken through $B$ and $C$, intersect the lines $AC$ and $AB$, respectively at the points $Q$ and $R$. Prove that the points $P, Q$ ¸ and $R$ are collinear.
Geometry Mathley 2011-12, 16.2
Let $ABCD$ be a quadrilateral and $P$ a point in the plane of the quadrilateral. Let $M,N$ be on the sides $AC,BD$ respectively such that $PM \parallel BC, PN \parallel AD$. $AC$ meets $BD$ at $E$. Prove that the orthocenter of triangles $EBC, EAD, EMN$ are collinear if and only if $P$ is on the line $AB$.
Đỗ Thanh Sơn
PS. Instead of the word [b]collinear[/b], it was written [b]concurrent[/b], probably a typo.
2012 Singapore Junior Math Olympiad, 3
In $\vartriangle ABC$, the external bisectors of $\angle A$ and $\angle B$ meet at a point $D$. Prove that the circumcentre of $\vartriangle ABD$ and the points $C, D$ lie on the same straight line.
2005 All-Russian Olympiad Regional Round, 8.4
Given an acute triangle $ABC$. Points $B'$ and $C'$ are symmetrical, respectively, to vertices $B$ and $ C$ wrt straight lines $AC$ and $AB$. Let $P$ be the intersection point of the circumcircles of triangles $ABB'$ and $ACC'$, different from $A$. Prove that the center of the circumcircle of triangle $ABC$ lies on line $PA$.
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.
Geometry Mathley 2011-12, 3.3
A triangle $ABC$ is inscribed in circle $(O)$. $P1, P2$ are two points in the plane of the triangle. $P_1A, P_1B, P_1C$ meet $(O)$ again at $A_1,B_1,C_1$ . $P_2A, P_2B, P_2C$ meet $(O)$ again at $A_2,B_2,C_2$.
a) $A_1A_2, B_1B_2, C_1C_2$ intersect $BC,CA,AB$ at $A_3,B_3,C_3$. Prove that three points $A_3,B_3,C_3$ are collinear.
b) $P$ is a point on the line $P_1P_2. A_1P,B_1P,C_1P$ meet (O) again at $A_4,B_4,C_4$. Prove that three lines $A_2A_4,B_2B_4,C_2C_4$ are concurrent.
Trần Quang Hùng
2016 Costa Rica - Final Round, G1
Let $\vartriangle ABC$ be acute with orthocenter $H$. Let $X$ be a point on $BC$ such that $B-X-C$. Let $\Gamma$ be the circumscribed circle of $\vartriangle BHX$ and $\Gamma_2$ be the circumscribed circle of $\vartriangle CHX$. Let $E$ be the intersection of $AB$ with $\Gamma$ , and $D$ be the intersection of $AC$ with $\Gamma_2$. Let $L$ be the intersection of line $HD$ with $\Gamma$ and $J$ be the intersection of line $EH$ with $\Gamma_2$. Prove that points $L$, $X$, and $J$ are collinear.
Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.
Mathley 2014-15, 1
Let $AD, BE, CF$ be segments whose midpoints are on the same line $\ell$. The points $X, Y, Z$ lie on the lines $EF, FD, DE$ respectively such that $AX \parallel BY \parallel CZ \parallel \ell$. Prove that $X, Y, Z$ are collinear.
Tran Quang Hung, High School of Natural Sciences, Hanoi National University
2019 Junior Balkan Team Selection Tests - Romania, 3
A circle with center $O$ is internally tangent to two circles inside it at points $S$ and $T$. Suppose the two circles inside intersect at $M$ and $N$ with $N$ closer to $ST$. Show that $OM$ and $MN$ are perpendicular if and only if $S,N, T$ are collinear.
2008 Oral Moscow Geometry Olympiad, 6
Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$.
(A. Zaslavsky)
2017-IMOC, G4
Given an acute $\vartriangle ABC$ with orthocenter $H$. Let $M_a$ be the midpoint of $BC. M_aH$ intersects the circumcircle of $\vartriangle ABC$ at $X_a$ and $AX_a$ intersects $BC$ at $Y_a$. Define $Y_b, Y_c$ in a similar way. Prove that $Y_a, Y_b,Y_c$ are collinear.
[img]https://2.bp.blogspot.com/-yjISBHtRa0s/XnSKTrhhczI/AAAAAAAALds/e_rvs9glp60L1DastlvT0pRFyP7GnJnCwCK4BGAYYCw/s320/imoc2017%2Bg4.png[/img]
2007 Thailand Mathematical Olympiad, 3
Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.
2004 Tournament Of Towns, 4
Two circles intersect in points $A$ and $B$. Their common tangent nearer $B$ touches the circles at points $E$ and $F$, and intersects the extension of $AB$ at the point $M$. The point $K$ is chosen on the extention of $AM$ so that $KM = MA$. The line $KE$ intersects the circle containing $E$ again at the point $C$. The line $KF$ intersects the circle containing $F$ again at the point $D$. Prove that the points $A, C$ and $D$ are collinear.
2018 Czech-Polish-Slovak Junior Match, 4
A line passing through the center $M$ of the equilateral triangle $ABC$ intersects sides $BC$ and $CA$, respectively, in points $D$ and $E$. Circumcircles of triangle $AEM$ and $BDM$ intersects, besides point $M$, also at point $P$. Prove that the center of circumcircle of triangle $DEP$ lies on the perpendicular bisector of the segment $AB$.
Novosibirsk Oral Geo Oly VII, 2019.4
Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear.
[img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]
1990 Tournament Of Towns, (250) 4
Let $ABCD$ be a rhombus and $P$ be a point on its side $BC$. The circle passing through $A, B$, and $P$ intersects $BD$ once more at the point $Q$ and the circle passing through $C,P$ and $Q$ intersects $BD$ once more at the point $R$. Prove that $A, R$ and $P$ lie on the one straight line.
(D. Fomin, Leningrad)