Found problems: 254
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.
1991 Greece National Olympiad, 2
Given two circles $(C_1)$ and $(C_2)$ with centers $\displaystyle{O_1}$ and $O_2$ respectively, intersecting at points $A$ and $B$. Let $AC$ και $AD$ be the diameters of $(C_1)$ and $(C_2)$ respectively . Tangent line of circle $(C_1)$ at point $A$ intersects $(C_2)$ at point $M$ and tangent line of circle $(C_2)$ at point A intersects $(C_1)$ at point $N$. Let $P$ be a point on line $AB$ such that $AB=BP$. Prove that:
a) Points $B,C,D$ are collinear.
b) Quadrilateral $AMPN$ is cyclic.
1997 Moldova Team Selection Test, 7
Let $ABC$ be a triangle with orthocenter $H$. Let the circle $\omega$ have $BC$ as the diameter. Draw tangents $AP$, $AQ$ to the circle $\omega $ at the point $P, Q$ respectively. Prove that $ P,H,Q$ lie on the same line .
2008 District Olympiad, 3
Let $ABCDA' B' C' D '$ be a cube , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.
2022-IMOC, G2
The incenter of triangle $ABC$ is $ I$. the circumcircle of $ABC$ is tangent to $BC$, $CA$, $AB$ at $T, E, F$. $R$ is a point on $BC$ . Let the $C$-excenter of $\vartriangle CER$ be $L$. Prove that points $L,T,F$ are collinear if and only if $B,E,A,R$ are concyclic.
[i]proposed by kyou46[/i]
2001 BAMO, 2
Let $JHIZ$ be a rectangle, and let $A$ and $C$ be points on sides $ZI$ and $ZJ,$ respectively. The perpendicular from $A$ to $CH$ intersects line $HI$ in $X$ and the perpendicular from $C$ to $AH$ intersects line $HJ$ in $Y.$ Prove that $X,$ $Y,$ and $Z$ are collinear (lie on the same line).
2024 Alborz Mathematical Olympiad, P4
In triangle \( ABC \), let \( I \) be the \( A \)-excenter. Points \( X \) and \( Y \) are placed on line \( BC \) such that \( B \) is between \( X \) and \( C \), and \( C \) is between \( Y \) and \( B \). Moreover, \( B \) and \( C \) are the contact points of \( BC \) with the \( A \)-excircle of triangles \( BAY \) and \( AXC \), respectively. Let \( J \) be the \( A \)-excenter of triangle \( AXY \), and let \( H' \) be the reflection of the orthocenter of triangle \( ABC \) with respect to its circumcenter. Prove that \( I \), \( J \), and \( H' \) are collinear.
Proposed by Ali Nazarboland
1972 Vietnam National Olympiad, 3
$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.
2000 All-Russian Olympiad Regional Round, 11.6
A circle inscribed in triangle $ABC$ has center $O$ and touches side $AC$ at point $K$. A second circle also has center $O$, intersects all sides of triangle $ABC$. Let $E$ and $F$ be the corresponding points of intersection with sides $AB$ and $BC$, closest to vertex $B$; $B_1$ and $B_2$ are the points of its intersection with side $AC$, and $B_1$ is closer to $A$. Prove that points $B$, $K$ and point $P$, the intersections of the segments $B_2E$ and $B_1F$ lie on the same straight line.
2022 Yasinsky Geometry Olympiad, 4
In the triangle $ABC$ the relationship $AB+AC = 2BC$ holds. Let $I$ and $M$ be the incenter and intersection point of the medians of triangle $ABC$ respectively, $AL$ its angle bisector, and point $P$ the orthocenter of triangle $BIC$. Prove that the points $L, M, P$ lie on a straight line.
(Matvii Kurskyi)
2023 Austrian Junior Regional 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]
2022-IMOC, G4
Let $\vartriangle ABC$ be an acute triangle with circumcircle $\Omega$. A line passing through $A$ perpendicular to $BC$ meets $\Omega$ again at $D$. Draw two circles $\omega_b$, $\omega_c$ with $B, C$ as centers and $BD$, $CD$ as radii, respectively, and they intersect $AB$, $AC$ at $E, F,$ respectively. Let $K\ne A$ be the second intersection of $(AEF)$ and $\Omega$, and let $\omega_b$, $\omega_c$ intersect $KB$, $KC$ at $P, Q$, respectively. The circumcenter of triangle $DP Q$ is $O$, prove that $K, O, D$ are collinear.
[i]proposed by Li4[/i]
Geometry Mathley 2011-12, 9.1
Let $ABC$ be a triangle with $(O), (I)$ being the circumcircle, and incircle respectively. Let $(I)$ touch $BC,CA$, and $AB$ at $A_0, B_0, C_0$ let $BC,CA$, and $AB$ intersect $B_0C_0, C_0A_0, A_0Bv$ at $A_1, B_1$, and $C_1$ respectively. Prove that $OI$ passes through the orthocenter of four triangles $A_0B_0C_0, A_0B_1C_1, B_0C_1A_1,C_0A_1B_1$.
Nguyễn Minh Hà
2012 Junior Balkan Team Selection Tests - Romania, 4
The quadrilateral $ABCD$ is inscribed in a circle centered at $O$, and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$. Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$.
a) Prove that the points $P, Q$, and $R$ are collinear.
b) If $U$ and $V$ are the circumcenters of the triangles $ABP$, and $CDP$, respectively, prove that the points $U, R, O, V$ are concyclic.
1960 Poland - Second Round, 3
There are two circles with a common center $ O $ and a point $ A $. Construct a circle with center $ A $ intersecting the given circles at points $ M $ and $ N $ such that the line $ MN $ passes through point $ O $.
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.
2015 Singapore Senior Math Olympiad, 1
In an acute-angled triangle $ABC$, $M$ is a point on the side $BC$, the line $AM$ meets the circumcircle $\omega$ of $ABC$ at the point $Q$ distinct from $A$. The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to the diameter $AK$ of $\omega$ at the point $P$. Let $L$ be the point on $\omega$ distinct from $Q$ such that $PL$ is tangent to $\omega$ at $L$. Prove that $L,M$ and $K$ are collinear.
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.
1985 Bundeswettbewerb Mathematik, 2
Prove that in every triangle for each of its altitudes: If you project the foof of one altitude on the other two altitudes and on the other two sides of the triangle, those four projections lie on the same line.
Geometry Mathley 2011-12, 9.2
Let $ABDE, BCFZ$ and $CAKL$ be three arbitrary rectangles constructed outside a triangle $ABC$. Let $EF$ meet $ZK$ at $M$, and $N$ be the intersection of the lines through $F,Z$ perpendicular to $FL,ZD$. Prove that $A,M,N$ are collinear.
Kostas Vittas
Geometry Mathley 2011-12, 9.3
Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$.
(a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$.
(b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear.
Nguyễn Văn Linh
2020 Thailand TSTST, 3
Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.
Indonesia Regional MO OSP SMA - geometry, 2020.4
It is known that triangle $ABC$ is not isosceles with altitudes of $AA_1, BB_1$, and $CC_1$. Suppose $B_A$ and $C_A$ respectively points on $BB_1$ and $CC_1$ so that $A_1B_A$ is perpendicular on $BB_1$ and $A_1C_A$ is perpendicular on $CC_1$. Lines $B_AC_A$ and $BC$ intersect at the point $T_A$ . Define in the same way the points $T_B$ and $T_C$ . Prove that points $T_A, T_B$, and $T_C$ are collinear.
2022 EGMO, 1
Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.
Russian TST 2017, P3
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.