Found problems: 478
2016 CentroAmerican, 2
Let $ABC$ be an acute-angled triangle, $\Gamma$ its circumcircle and $M$ the midpoint of $BC$. Let $N$ be a point in the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle NAC= \angle BAM$. Let $R$ be the midpoint of $AM$, $S$ the midpoint of $AN$ and $T$ the foot of the altitude through $A$. Prove that $R$, $S$ and $T$ are collinear.
2003 Oral Moscow Geometry Olympiad, 5
Given triangle $ABC$. Point $O_1$ is the center of the $BCDE$ rectangle, constructed so that the side $DE$ of the rectangle contains the vertex $A$ of the triangle. Points $O_2$ and $O_3$ are the centers of rectangles constructed in the same way on the sides $AC$ and $AB$, respectively. Prove that lines $AO_1, BO_2$ and $CO_3$ meet at one point.
Fractal Edition 2, P4
In triangle $ABC$, the points $D$, $E$, and $F$ are the feet of the perpendiculars dropped from $A$, $B$, and $C$, respectively, onto the opposite sides. The point $X_A$ is such that a circle passing through $E$ and $F$ is tangent to the circumcircle of triangle $ABC$ at $X_A$, and $X_A$ is on a different side of $EF$ as $A$. Similarly, $X_B$ and $X_C$ are defined. Prove that the lines $AX_A$, $BX_B$, and $CX_C$ are concurrent.
2022 JBMO Shortlist, G4
Given is an equilateral triangle $ABC$ and an arbitrary point, denoted by $E$, on the line segment $BC$. Let $l$ be the line through $A$ parallel to $BC$ and let $K$ be the point on $l$ such that $KE$ is perpendicular to $BC$. The circle with centre $K$ and radius $KE$ intersects the sides $AB$ and $AC$ at $M$ and $N$, respectively. The line perpendicular to $AB$ at $M$ intersects $l$ at $D$, and the line perpendicular to $AC$ at $N$ intersects $l$ at $F$. Show that the point of intersection of the angle bisectors of angles $MDA$ and $NFA$ belongs to the line $KE$.
2008 Switzerland - Final Round, 8
Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$
2015 Saudi Arabia GMO TST, 3
Let $ABC$ be a triangle and $G$ its centroid. Let $G_a, G_b$ and $G_c$ be the orthogonal projections of $G$ on sides $BC, CA$, respectively $AB$. If $S_a, S_b$ and $S_c$ are the symmetrical points of $G_a, G_b$, respectively $G_c$ with respect to $G$, prove that $AS_a, BS_b$ and $CS_c$ are concurrent.
Liana Topan
2023 Indonesia TST, G
Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$
2017 Saudi Arabia JBMO TST, 4
Let $ABC$ be an acute, non isosceles triangle and $(O)$ be its circumcircle (with center $O$). Denote by $G$ the centroid of the triangle $ABC$, by $H$ the foot of the altitude from $A$ onto the side $BC$ and by $I$ the midpoint of $AH$. The line $IG$ intersects $BC$ at $K$.
1. Prove that $CK = BH$.
2. The ray $(GH$ intersects $(O)$ at L. Denote by $T$ the circumcenter of the triangle $BHL$. Prove that $AO$ and $BT$ intersect on the circle $(O)$.
2015 Sharygin Geometry Olympiad, P13
Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$. Point $M$ is the midpoint of $H_2H_3$. Line $AM$ meets $H_2H_1$ at point $K$. Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$.
2020 Federal Competition For Advanced Students, P2, 1
Let $ABCD$ be a convex cyclic quadrilateral with the diagonal intersection $S$. Let further be $P$ the circumcenter of the triangle $ABS$ and $Q$ the circumcenter of the triangle $BCS$. The parallel to $AD$ through $P$ and the parallel to $CD$ through $Q$ intersect at point $R$. Prove that $R$ is on $BD$.
(Karl Czakler)
2023 Romanian Master of Mathematics Shortlist, G1
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. The incircle of the triangle $ABC$
touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. The circumcircle of triangle $ADI$ crosses $\omega$ again at $P$, and the lines $PE$ and $PF$ cross $\omega$ again at $X$and $Y$, respectively. Prove that the lines $AI$, $BX$ and $CY$ are concurrent.
2012 Sharygin Geometry Olympiad, 2
Three parallel lines passing through the vertices $A, B$, and $C$ of triangle $ABC$ meet its circumcircle again at points $A_1, B_1$, and $C_1$ respectively. Points $A_2, B_2$, and $C_2$ are the reflections of points $A_1, B_1$, and $C_1$ in $BC, CA$, and $AB$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ are concurrent.
(D.Shvetsov, A.Zaslavsky)
1980 IMO Shortlist, 17
Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.
Swiss NMO - geometry, 2018.4
Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.
2018 IMO Shortlist, G4
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
1979 Bundeswettbewerb Mathematik, 2
The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.
2021 Peru Iberoamerican Team Selection Test, P4
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
1985 Tournament Of Towns, (080) T1
A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .
1988 IMO Shortlist, 15
Let $ ABC$ be an acute-angled triangle. The lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are constructed through the vertices $ A$, $ B$ and $ C$ respectively according the following prescription: Let $ H$ be the foot of the altitude drawn from the vertex $ A$ to the side $ BC$; let $ S_{A}$ be the circle with diameter $ AH$; let $ S_{A}$ meet the sides $ AB$ and $ AC$ at $ M$ and $ N$ respectively, where $ M$ and $ N$ are distinct from $ A$; then let $ L_{A}$ be the line through $ A$ perpendicular to $ MN$. The lines $ L_{B}$ and $ L_{C}$ are constructed similarly. Prove that the lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are concurrent.
Geometry Mathley 2011-12, 2.3
Let $ABC$ be a triagle inscribed in a circle $(O)$. A variable line through the orthocenter $H$ of the triangle meets the circle $(O)$ at two points $P , Q$. Two lines through $P, Q$ that are perpendicular to $AP , AQ$ respectively meet $BC$ at $M, N$ respectively. Prove that the line through $P$ perpendicular to $OM$ and the line through $Q$ perpendicular to $ON$ meet each other at a point on the circle $(O)$.
Nguyễn Văn Linh
2022 IFYM, Sozopol, 2
Let $k$ be the circumcircle of the acute triangle $ABC$. Its inscribed circle touches sides $BC$, $CA$ and $AB$ at points $D, E$ and $F$ respectively. The line $ED$ intersects $k$ at the points $M$ and $N$, so that $E$ lies between $M$ and $D$. Let $K$ and $L$ be the second intersection points of the lines $NF$ and $MF$ respectively with $k$. Let $AK \cap BL = Q$. Prove that the lines $AL$, $BK$ and $QF$ intersect at a point.
1963 All Russian Mathematical Olympiad, 027
Given $5$ circumferences, every four of them have a common point. Prove that there exists a point that belongs to all five circumferences.
2014 IMAC Arhimede, 2
A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.
2019 Durer Math Competition Finals, 5
Let $ABC$ be an acute triangle and let $X, Y , Z$ denote the midpoints of the shorter arcs $BC, CA, AB$ of its circumcircle, respectively. Let $M$ be an arbitrary point on side $BC$. The line through $M$, parallel to the inner angular bisector of $\angle CBA$ meets the outer angular bisector of $\angle BCA$ at point $N$. The line through $M$, parallel to the inner angular bisector of $\angle BCA$ meets the outer angular bisector of $\angle CBA$ at point $P$. Prove that lines $XM, Y N, ZP$ pass through a single point.
1984 All Soviet Union Mathematical Olympiad, 378
The circle with the centre $O$ is inscribed in the triangle $ABC$ . The circumference touches its sides $[BC], [CA], [AB]$ in $A_1, B_1, C_1$ points respectively. The $[AO], [BO], [CO]$ segments cross the circumference in $A_2, B_2, C_2$ points respectively. Prove that lines $(A_1A_2),(B_1B_2)$ and $(C_1C_2)$ intersect in one point.