Found problems: 670
2019 Yasinsky Geometry Olympiad, p4
Find the angles of the cyclic quadrilateral if you know that each of its diagonals is a bisector of one angle and a trisector of the opposite one (the trisector of the angle is one of the two rays that lie in the interior of the angle and divide it into three equal parts).
(Vyacheslav Yasinsky)
Indonesia MO Shortlist - geometry, g9
It is known that $ABCD$ is a parallelogram. The point $E$ is taken so that $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line that passes through $A$, intersects the segment $DC$ at point $F$ and intersects the extension of the line $BC$ at $G$. Given $EF = EG = EC$. Prove that $\ell$ is the bisector of the angle $\angle BAD$.
2016 China Team Selection Test, 6
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.
2008 International Zhautykov Olympiad, 2
Let $ A_1A_2$ be the external tangent line to the nonintersecting cirlces $ \omega_1(O_1)$ and $ \omega_2(O_2)$,$ A_1\in\omega_1$,$ A_2\in\omega_2$.Points $ K$ is the midpoint of $ A_1A_2$.And $ KB_1$ and $ KB_2$ are tangent lines to $ \omega_1$ and $ \omega_2$,respectvely($ B_1\neq A_1$,$ B_2\neq A_2$).Lines $ A_1B_1$ and $ A_2B_2$ meet in point $ L$,and lines $ KL$ and $ O_1O_2$ meet in point $ P$.
Prove that points $ B_1,B_2,P$ and $ L$ are concyclic.
OIFMAT I 2010, 2
In an acute angle $ \vartriangle ABC $, let $ AD, BE, CF $ be their altitudes (with $ D, E, F $ lying on $ BC, CA, AB $, respectively). Let's call $ O, H $ the circumcenter and orthocenter of $ \vartriangle ABC $, respectively. Let $ P = CF \cap AO $. Suppose the following two conditions are true:
$\bullet$ $ FP = EH $
$\bullet$ There is a circle that passes through points $ A, O, H, C $
Prove that the $ \vartriangle ABC $ is equilateral.
2006 Romania Team Selection Test, 2
Let $A$ be point in the exterior of the circle $\mathcal C$. Two lines passing through $A$ intersect the circle $\mathcal C$ in points $B$ and $C$ (with $B$ between $A$ and $C$) respectively in $D$ and $E$ (with $D$ between $A$ and $E$). The parallel from $D$ to $BC$ intersects the second time the circle $\mathcal C$ in $F$. Let $G$ be the second point of intersection between the circle $\mathcal C$ and the line $AF$ and $M$ the point in which the lines $AB$ and $EG$ intersect. Prove that
\[ \frac 1{AM} = \frac 1{AB} + \frac 1{AC}. \]
2024 Bangladesh Mathematical Olympiad, P9
Let $ABC$ be a triangle and $M$ be the midpoint of side $BC$. The perpendicular bisector of $BC$ intersects the circumcircle of $\triangle ABC$ at points $K$ and $L$ ($K$ and $A$ lie on the opposite sides of $BC$). A circle passing through $L$ and $M$ intersects $AK$ at points $P$ and $Q$ ($P$ lies on the line segment $AQ$). $LQ$ intersects the circumcircle of $\triangle KMQ$ again at $R$. Prove that $BPCR$ is a cyclic quadrilateral.
2012 India IMO Training Camp, 1
The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.
2025 NEPALTST, 3
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.
Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.
If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.
$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
2018 USAMTS Problems, 3:
Cyclic quadrilateral $ABCD$ has $AC\perp BD$, $AB+CD=12$, and $BC+AD=13$. FInd the greatest possible area of $ABCD$.
2019 Dutch IMO TST, 1
Let $ABCD$ be a cyclic quadrilateral (In the same order) inscribed into the circle $\odot (O)$. Let $\overline{AC}$ $\cap$ $\overline{BD}$ $=$ $E$. A randome line $\ell$ through $E$ intersects $\overline{AB}$ at $P$ and $BC$ at $Q$. A circle $\omega$ touches $\ell$ at $E$ and passes through $D$. Given, $\omega$ $\cap$ $\odot (O)$ $=$ $R$. Prove, Points $B,Q,R,P$ are concyclic.
2017 Estonia Team Selection Test, 3
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2023 Romania EGMO TST, P3
In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .
1996 APMO, 3
If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.
2004 National Olympiad First Round, 29
Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral?
$
\textbf{(A)}\ 5\sqrt 3
\qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3}
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \sqrt{34}
$
1985 Brazil National Olympiad, 3
A convex quadrilateral is inscribed in a circle of radius $1$. Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$.
2022 Thailand TST, 2
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.
2023 Indonesia TST, 1
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
2021 Saudi Arabia Training Tests, 21
Let $ABCD$ be a cyclic quadrilateral with $O$ is circumcenter and $AC$ meets $BD$ at $I$ Suppose that rays $DA,CD$ meet at $E$ and rays $BA,CD$ meet at $F$. The Gauss line of $ABCD$ meets $AB,BC,CD,DA$ at points $M,N,P,Q$ respectively. Prove that the circle of diameter $OI$ is tangent to two circles $(ENQ), (FMP)$
2016 All-Russian Olympiad, 2
Diagonals $AC,BD$ of cyclic quadrilateral $ABCD$ intersect at $P$.Point $Q$ is on$BC$ (between$B$ and $C$) such that $PQ \perp AC$.Prove that the line passes through the circumcenters of triangles $APD$ and $BQD$ is parallel to $AD$.(A.Kuznetsov)