Olympiad problems

Geometry Mathley 2011-12, 16.1

Let $ABCD$ be a cyclic quadrilateral with two diagonals intersect at $E$. Let $ M$, $N$, $P$, $Q$ be the reflections of $ E $ in midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Prove that the Euler lines of $ \triangle MAB$, $\triangle NBC$, $\triangle PCD,$ $\triangle QDA$ are concurrent. Trần Quang Hùng

2014 Oral Moscow Geometry Olympiad, 4

Tags: geometry , euler line , collinear , centroid , orthocenter , perpendicular

The medians $AA_0, BB_0$, and $CC_0$ of the acute-angled triangle $ABC$ intersect at the point $M$, and heights $AA_1, BB_1$ and $CC_1$ at point $H$. Tangent to the circumscribed circle of triangle $A_1B_1C_1$ at $C_1$ intersects the line $A_0B_0$ at the point $C'$. Points $A'$ and $B'$ are defined similarly. Prove that $A', B'$ and $C'$ lie on one line perpendicular to the line $MH$.

2020 Sharygin Geometry Olympiad, 9

Tags: euler line , geometry

The vertex $A$, center $O$ and Euler line $\ell$ of a triangle $ABC$ is given. It is known that $\ell$ intersects $AB,AC$ at two points equidistant from $A$. Restore the triangle.

2020 Romanian Master of Mathematics Shortlist, G2

Tags: geometry , euler line

Let $ABC$ be an acute scalene triangle, and let $A_1, B_1, C_1$ be the feet of the altitudes from $A, B, C$. Let $A_2$ be the intersection of the tangents to the circle $ABC$ at $B, C$ and define $B_2, C_2$ similarly. Let $A_2A_1$ intersect the circle $A_2B_2C_2$ again at $A_3$ and define $B_3, C_3$ similarly. Show that the circles $AA_1A_3, BB_1B_3$, and $CC_1C_3$ all have two common points, $X_1$ and $X_2$ which both lie on the Euler line of the triangle $ABC$. [i]United Kingdom, Joe Benton[/i]

1999 Estonia National Olympiad, 3

Tags: euler line , centroid , orthocenter , parallel , geometry , trigonometry

Prove that the line segment, joining the orthocenter and the intersection point of the medians of the acute-angled triangle $ABC$ is parallel to the side $AB$ iff $\tan \angle A \cdot \tan \angle B = 3$.

2020-IMOC, G3

Tags: geometry , circumcircle , euler line , incenter

Triangle $ABC$ has incenter $I$ and circumcenter $O$. $AI, BI, CI$ intersect the circumcircle of $ABC$ again at $M_A, M_B, M_C$, respectively. Show that the Euler line of $BIC$ passes through the circumcenter of $OM_BM_C$. (houkai)

2016 Sharygin Geometry Olympiad, P16

Tags: geometry , circumcircle , euler line

Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$. The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$.

Kvant 2024, M2781

Tags: geometry , euler line

Let $A_1$ be the midpoint of the smaller arc $BC$ of the circumcircle of the acute-angled triangle $ABC.{}$ The point $A_1$ is reflected relative to the side $BC,$ and then its image is reflected relative to the bisector of $\angle BAC{}$ resulting in the point $A_2 $. Similarly, the points $B_2$ and $C_2$ are constructed. Prove that the circumcenter and incenter of the triangle $ABC{}$ lie on the Euler line of the triangle $A_2B_2C_2.$ [i]Proposed by A. Tereshin[/i]

2025 Sharygin Geometry Olympiad, 9

Tags: geometry , euler line , thalestheorem

The line $l$ passing through the orthocenter $H$ of a triangle $ABC$ $(BC>AB)$ and parallel to $AC$ meets $AB$ and $BC$ at points $D$ and $E$ respectively. The line passing through the circumcenter of the triangle and parallel to the median $BM$ meets $l$ at point $F$. Prove that the length of segment $HF$ is three times greater than the difference of $FE$ and $DH$ Proposed by: A.Mardanov, K.Mardanova

2021 Iran Team Selection Test, 6

Tags: geometry , euler line

Point $D$ is chosen on the Euler line of triangle $ABC$ and it is inside of the triangle. Points $E,F$ are were the line $BD,CD$ intersect with $AC,AB$ respectively. Point $X$ is on the line $AD$ such that $\angle EXF =180 - \angle A$, also $A,X$ are on the same side of $EF$. If $P$ is the second intersection of circumcircles of $CXF,BXE$ then prove the lines $XP,EF$ meet on the altitude of $A$ Proposed by [i]Alireza Danaie[/i]

2019 Belarus Team Selection Test, 2.2

Tags: circumcircle , perpendicular bisector , nine point center , euler line , geometry

Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$. [i](M. Berindeanu, RMC 2018 book)[/i]