Found problems: 3882
2004 Postal Coaching, 11
Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.
2005 MOP Homework, 3
Points $M$ and $M'$ are isogonal conjugates in the traingle $ABC$. We draw perpendiculars $MP$, $MQ$, $MR$, and $M'P'$, $M'Q'$, $M'R'$ to the sides $BC$, $AC$, $AB$ respectively. Let $QR$, $Q'R'$, and $RP$, $R'P'$ and $PQ$, $P'Q'$ intersect at $E$, $F$, $G$ respectively. Show that the lines $EA$, $FB$, and $GC$ are parallel.
2019 Azerbaijan IMO TST, 2
Let $ABC$ ($BC > AC$) be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $AB$ at the point $D$. The circumcircles of triangles $BCD, OCD$ and $AOB$ intersect the ray $CA$ (beyond $A$) at the points $Q, P$ and $K$, respectively, such that $P \in (AK)$ and $K \in (PQ)$. The line $PD$ intersects the circumcircle of triangle $BKQ$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $BQ$. Prove that $TB = TQ$.
2003 Finnish National High School Mathematics Competition, 1
The incentre of the triangle $ABC$ is $I.$ The rays $AI, BI$ and $CI$ intersect the circumcircle of the triangle $ABC$ at the points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.
2008 Harvard-MIT Mathematics Tournament, 6
In a triangle $ ABC$, take point $ D$ on $ BC$ such that $ DB \equal{} 14, DA \equal{} 13, DC \equal{} 4$, and the circumcircle of $ ADB$ is congruent to the circumcircle of $ ADC$. What is the area of triangle $ ABC$?
2011 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$
1995 IMO Shortlist, 8
Suppose that $ ABCD$ is a cyclic quadrilateral. Let $ E \equal{} AC\cap BD$ and $ F \equal{} AB\cap CD$. Denote by $ H_{1}$ and $ H_{2}$ the orthocenters of triangles $ EAD$ and $ EBC$, respectively. Prove that the points $ F$, $ H_{1}$, $ H_{2}$ are collinear.
Original formulation:
Let $ ABC$ be a triangle. A circle passing through $ B$ and $ C$ intersects the sides $ AB$ and $ AC$ again at $ C'$ and $ B',$ respectively. Prove that $ BB'$, $CC'$ and $ HH'$ are concurrent, where $ H$ and $ H'$ are the orthocentres of triangles $ ABC$ and $ AB'C'$ respectively.
2007 India IMO Training Camp, 1
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
2025 Romania National Olympiad, 2
Let $\triangle ABC$ be an acute-angled triangle, with circumcenter $O$, circumradius $R$ and orthocenter $H$. Let $A_1$ be a point on $BC$ such that $A_1H+A_1O=R$. Define $B_1$ and $C_1$ similarly.
If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$, prove that $\triangle ABC$ is equilateral.
2008 USA Team Selection Test, 7
Let $ ABC$ be a triangle with $ G$ as its centroid. Let $ P$ be a variable point on segment $ BC$. Points $ Q$ and $ R$ lie on sides $ AC$ and $ AB$ respectively, such that $ PQ \parallel AB$ and $ PR \parallel AC$. Prove that, as $ P$ varies along segment $ BC$, the circumcircle of triangle $ AQR$ passes through a fixed point $ X$ such that $ \angle BAG = \angle CAX$.
2005 Iran MO (3rd Round), 2
Suppose $O$ is circumcenter of triangle $ABC$. Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$. Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is \[\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}\]
2022 Turkey Junior National Olympiad, 4
In parallellogram $ABCD$, on the arc $BC$ of the circumcircle $(ABC)$, not containing the point $A$, we take a point $P$ and on the $[AC$, we take a point $Q$ such that $\angle PBC= \angle CDQ$. Prove that $(APQ)$ is tangent to $AB$.
2015 Taiwan TST Round 3, 2
Let $O$ be the circumcircle of the triangle $ABC$. Two circles $O_1,O_2$ are tangent to each of the circle $O$ and the rays $\overrightarrow{AB},\overrightarrow{AC}$, with $O_1$ interior to $O$, $O_2$ exterior to $O$. The common tangent of $O,O_1$ and the common tangent of $O,O_2$ intersect at the point $X$. Let $M$ be the midpoint of the arc $BC$ (not containing the point $A$) on the circle $O$, and the segment $\overline{AA'}$ be the diameter of $O$. Prove that $X,M$, and $A'$ are collinear.
2016 Sharygin Geometry Olympiad, 7
Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point.
[i](The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)[/i]
2007 Baltic Way, 14
In a convex quadrilateral $ABCD$ we have $ADC = 90^{\circ}$. Let $E$ and $F$ be the projections of $B$ onto the lines $AD$ and $AC$, respectively. Assume that $F$ lies between $A$ and $C$, that $A$ lies between $D$ and $E$, and that the line $EF$ passes through the midpoint of the segment $BD$. Prove that the quadrilateral $ABCD$ is cyclic.
2016 Mediterranean Mathematics Olympiad, 1
Let $ABC$ be a triangle. Let $D$ be the intersection point of the angle bisector at $A$ with $BC$.
Let $T$ be the intersection point of the tangent line to the circumcircle of triangle $ABC$ at point $A$ with the line through $B$ and $C$.
Let $I$ be the intersection point of the orthogonal line to $AT$ through point $D$ with the altitude $h_a$ of the triangle at point $A$.
Let $P$ be the midpoint of $AB$, and let $O$ be the circumcenter of triangle $ABC$.
Let $M$ be the intersection point of $AB$ and $TI$, and let $F$ be the intersection point of $PT$ and $AD$.
Prove: $MF$ and $AO$ are orthogonal to each other.
2014 India Regional Mathematical Olympiad, 5
Let $ABC$ be an acute angled triangle with $H$ as its orthocentre. For any point $P$ on the circumcircle of triangle $ABC$, let $Q$ be the point of intersection of the line $BH$ with line $AP$. Show that there is a unique point $X$ on the circumcircle of triangle $ABC$ such that for every $P$ other than $B,C$, the circumcircle of $HPQ$ passes through $X$.
2017 CentroAmerican, 3
Let $ABC$ be a triangle and $D$ be the foot of the altitude from $A$. Let $l$ be the line that passes through the midpoints of $BC$ and $AC$. $E$ is the reflection of $D$ over $l$. Prove that the circumcentre of $\triangle ABC$ lies on the line $AE$.
1971 IMO Longlists, 7
In a triangle $ABC$, let $H$ be its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Prove that:
[b](a)[/b] $|OH| = R \sqrt{1-8 \cos \alpha \cdot \cos \beta \cdot \cos \gamma}$ where $\alpha, \beta, \gamma$ are angles of the triangle $ABC;$
[b](b)[/b] $O \equiv H$ if and only if $ABC$ is equilateral.
1994 IMO Shortlist, 3
A circle $ C$ has two parallel tangents $ L'$ and$ L"$. A circle $ C'$ touches $ L'$ at $ A$ and $ C$ at $ X$. A circle $ C"$ touches $ L"$ at $ B$, $ C$ at $ Y$ and $ C'$ at $ Z$. The lines $ AY$ and $ BX$ meet at $ Q$. Show that $ Q$ is the circumcenter of $ XYZ$
1959 IMO Shortlist, 5
An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$.
a) Prove that $N$ and $N'$ coincide;
b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$;
c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.
2014 Indonesia MO, 2
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.
2023 Canadian Mathematical Olympiad Qualification, 6
Given triangle $ABC$ with circumcircle $\Gamma$, let $D$, $E$, and $F$ be the midpoints of sides $BC$, $CA$, and $AB$, respectively, and let the lines $AD$, $BE$, and $CF$ intersect $\Gamma$ again at points $J$, $K$, and $L$, respectively. Show that the area of triangle $JKL$ is at least that of triangle $ABC$.
2018 HMNT, 8
Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area $3$ and triangle $ACD$ has area $4$, find the area of triangle $ABC$.
2019 Iran MO (3rd Round), 3
Consider a triangle $ABC$ with circumcenter $O$ and incenter $I$. Incircle touches sides $BC,CA$ and $AB$ at $D, E$ and $F$. $K$ is a point such that $KF$ is tangent to circumcircle of $BFD$ and $KE$ is tangent to circumcircle of $CED$. Prove that $BC,OI$ and $AK$ are concurrent.