Found problems: 3882
2007 South East Mathematical Olympiad, 2
In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.
Kyiv City MO Seniors 2003+ geometry, 2017.11.5.1
The bisector $AD$ is drawn in the triangle $ABC$. Circle $k$ passes through the vertex $A$ and touches the side $BC$ at point $D$. Prove that the circle circumscribed around $ABC$ touches the circle $k$ at point $A$.
2024 Sharygin Geometry Olympiad, 15
The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.
2012 Online Math Open Problems, 16
Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$.
[i]Ray Li.[/i]
2018 China Western Mathematical Olympiad, 4
In acute angled $\triangle ABC$, $AB > AC$, points $E, F$ lie on $AC, AB$ respectively, satisfying $BF+CE = BC$. Let $I_B, I_C$ be the excenters of $\triangle ABC$ opposite $B, C$ respectively, $EI_C, FI_B$ intersect at $T$, and let $K$ be the midpoint of arc $BAC$. Let $KT$ intersect the circumcircle of $\triangle ABC$ at $K,P$. Show $T,F,P,E$ concyclic.
2013 ELMO Shortlist, 11
Let $\triangle ABC$ be a nondegenerate isosceles triangle with $AB=AC$, and let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively. $BE$ intersects the circumcircle of $\triangle ABC$ again at $G$, and $H$ is the midpoint of minor arc $BC$. $CF\cap DG=I, BI\cap AC=J$. Prove that $\angle BJH=\angle ADG$ if and only if $\angle BID=\angle GBC$.
[i]Proposed by David Stoner[/i]
1991 AIME Problems, 14
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
2016 Saint Petersburg Mathematical Olympiad, 5
Points $A$ and $P$ are marked in the plane not lying on the line $\ell$. For all right triangles $ABC$ with hypotenuse on $\ell$, show that the circumcircle of triangle $BPC$ passes through a fixed point other than $P$.
2016 Serbia National Math Olympiad, 3
Let $ABC$ be a triangle and $O$ its circumcentre. A line tangent to the circumcircle of the triangle $BOC$ intersects sides $AB$ at $D$ and $AC$ at $E$. Let $A'$ be the image of $A$ under $DE$. Prove that the circumcircle of the triangle $A'DE$ is tangent to the circumcircle of triangle $ABC$.
2017 Singapore MO Open, 1
The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.
2012 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
2000 All-Russian Olympiad, 3
Let $O$ be the center of the circumcircle $\omega$ of an acute-angle triangle $ABC$. A circle $\omega_1$ with center $K$ passes through $A$, $O$, $C$ and intersects $AB$ at $M$ and $BC$ at $N$. Point $L$ is symmetric to $K$ with respect to line $NM$. Prove that $BL \perp AC$.
2018 JBMO Shortlist, G3
Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.
1980 IMO Longlists, 10
Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.
2007 China Western Mathematical Olympiad, 2
Let $ C$ and $ D$ be two intersection points of circle $ O_1$ and circle $ O_2$. A line, passing through $ D$, intersects the circle $ O_1$ and the circle $ O_2$ at the points $ A$ and $ B$ respectively. The points $ P$ and $ Q$ are on circles $ O_1$ and $ O_2$ respectively. The lines $ PD$ and $ AC$ intersect at $ H$, and the lines $ QD$ and $ BC$ intersect at $ M$. Suppose that $ O$ is the circumcenter of the triangle $ ABC$. Prove that $ OD\perp MH$ if and only if $ P,Q,M$ and $ H$ are concyclic.
1970 IMO Longlists, 21
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
2011 China Second Round Olympiad, 1
Let $P,Q$ be the midpoints of diagonals $AC,BD$ in cyclic quadrilateral $ABCD$. If $\angle BPA=\angle DPA$, prove that $\angle AQB=\angle CQB$.
2019 Belarusian National Olympiad, 10.2
A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$.
Prove that the lines $BC$ and $B_1C_1$ are parallel.
[i](A. Voidelevich)[/i]
2012 CentroAmerican, 2
Let $\gamma$ be the circumcircle of the acute triangle $ABC$. Let $P$ be the midpoint of the minor arc $BC$. The parallel to $AB$ through $P$ cuts $BC, AC$ and $\gamma$ at points $R,S$ and $T$, respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$. Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$.
2017 Azerbaijan Team Selection Test, 1
Let $ABC$ be an acute angled triangle. Points $E$ and $F$ are chosen on the sides $AC$ and $AB$, respectively, such that \[BC^2=BA\times BF+CE\times CA.\] Prove that for all such $E$ and $F$, circumcircle of the triangle $AEF$ passes through a fixed point different from $A$.
2004 Baltic Way, 18
A ray emanating from the vertex $A$ of the triangle $ABC$ intersects the side $BC$ at $X$ and the circumcircle of triangle $ABC$ at $Y$. Prove that $\frac{1}{AX}+\frac{1}{XY}\geq \frac{4}{BC}$.
2009 Bundeswettbewerb Mathematik, 3
Given a triangle $ABC$ and a point $P$ on the side $AB$ .
Let $Q$ be the intersection of the straight line $CP$ (different from $C$) with the circumcicle of the triangle.
Prove the inequality $$\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2$$ and that equality holds if and only if the $CP$ is bisector of the angle $ACB$.
[img]https://cdn.artofproblemsolving.com/attachments/b/1/068fafd5564e77930160115a1cd409c4fdbf61.png[/img]
2005 USA Team Selection Test, 2
Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that
\[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]
Croatia MO (HMO) - geometry, 2014.7
Let point $I$ be the center of the inscribed circle of an acute-angled triangle $ABC$. Rays $AI$ and $BI$ intersect the circumcircle $k$ of triangle $ABC$ at points $D$ and $E$ respectively. The segments $DE$ and $CA$ intersect at point $F$, the line through point $E$ parallel to the line $FI$ intersects the circle $k$ at point $G$, and the lines $FI$ and $DG$ intersect at point $H$. Prove that the lines $CA$ and $BH$ touch the circumcircle of the triangle $DFH$ at the points $F$ and $H$ respectively.
Kharkiv City MO Seniors - geometry, 2018.10.4
On the sides $AB, AC ,BC$ of the triangle $ABC$, the points $M, N, K$ are selected, respectively, such that $AM = AN$ and $BM = BK$. The circle circumscribed around the triangle $MNK$ intersects the segments $AB$ and $BC$ for the second time at points $P$ and $Q$, respectively. Lines $MN$ and $PQ$ intersect at point $T$. Prove that the line $CT$ bisects the segment $MP$.