Found problems: 257
2000 All-Russian Olympiad Regional Round, 9.4
Circles $S_1$ and $S_2$ intersect at points $M$ and $N$. Through point $A$ of circle $S_1$, draw straight lines $AM$ and $AN$ intersecting $S_2$ at points $B$ and $C$, and through point $D$ of circle $S_2$, draw straight lines $DM$ and $DN$ intersecting $S_1$ at points $E$ and $F$, and $A$, $E$, $F$ lie along one side of line $MN$, and $D$, $B$, $C$ lie on the other side (see figure). Prove that if $AB = DE$, then points $A$, $F$, $C$ and $D$ lie on the same circle, the position of the center of which does not depend on choosing points $A$ and $D$.
[img]https://cdn.artofproblemsolving.com/attachments/7/0/d1f9c2f39352e2b39e55bd2538677073618ef9.png[/img]
2017 Czech And Slovak Olympiad III A, 5
Given is the acute triangle $ABC$ with the intersection of altitudes $H$. The angle bisector of angle $BHC$ intersects side $BC$ at point $D$. Mark $E$ and $F$ the symmetrics of the point $D$ wrt lines $AB$ and $AC$. Prove that the circle circumscribed around the triangle $AEF$ passes through the midpoint of the arc $BAC$
2017 JBMO Shortlist, G5
A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE, BCEF, CAFD, AEPF, BFPD$, and $CDPE$ are concyclic, then all six are concyclic.
2022 Saudi Arabia BMO + EGMO TST, 1.2
Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex $M$ from angle $CMD$. prove that $ABCD$ is a cyclc quadrilateral.
2018 Puerto Rico Team Selection Test, 2
Let $ABC$ be an acute triangle and let $P,Q$ be points on $BC$ such that $\angle QAC =\angle ABC$ and $\angle PAB = \angle ACB$. We extend $AP$ to $M$ so that $ P$ is the midpoint of $AM$ and we extend $AQ$ to $N$ so that $Q$ is the midpoint of $AN$. If T is the intersection point of $BM$ and $CN$, show that quadrilateral $ABTC$ is cyclic.
2001 Korea Junior Math Olympiad, 8
$ABCD$ is a convex quadrilateral, both $\angle ABC$ and $\angle BCD$ acute. $E$ is a point inside $ABCD$ satisfying $AE=DE$, and $X, Y$ are the intersection of $AD$ and $CE, BE$ respectively, but not $X=A$ or $Y=D$. If $ABEX$ and $CDEY$ are both inscribed quadrilaterals, prove that the distance between $E$ and the lines $AB, BC, CD$ are all equal.
2003 Estonia National Olympiad, 3
Let $ABC$ be a triangle and $A_1, B_1, C_1$ points on $BC, CA, AB$, respectively, such that the lines $AA_1, BB_1, CC_1$ meet at a single point. It is known that $A, B_1, A_1, B$ are concyclic and $B, C_1, B_1, C$ are concyclic. Prove that
a) $C, A_1, C_1, A$ are concyclic,
b) $AA_1,, BB_1, CC_1$ are the heights of $ABC$.
2021 Sharygin Geometry Olympiad, 10-11.5
A secant meets one circle at points $A_1$, $B_1$։, this secant meets a second circle at points $A_2$, $B_2$. Another secant meets the first circle at points $C_1$, $D_1$ and meets the second circle at points $C_2$, $D_2$. Prove that point $A_1C_1 \cap B_2D_2$, $A_1C_1 \cap A_2C_2$, $A_2C_2 \cap B_1D_1$, $B_2D_2 \cap B_1D_1$ lie on a circle coaxial with two given circles.
Geometry Mathley 2011-12, 13.3
Let $ABCD$ be a quadrilateral inscribed in circle $(O)$. Let $M,N$ be the midpoints of $AD,BC$. A line through the intersection $P$ of the two diagonals $AC,BD$ meets $AD,BC$ at $S, T$ respectively. Let $BS$ meet $AT$ at $Q$. Prove that three lines $AD,BC,PQ$ are concurrent if and only if $M, S, T,N$ are on the same circle.
Đỗ Thanh Sơn
1991 Mexico National Olympiad, 4
The diagonals $AC$ and $BD$ of a convex quarilateral $ABCD$ are orthogonal. Let $M,N,R,S$ be the midpoints of the sides $AB,BC,CD$ and $DA$ respectively, and let $W,X,Y,Z$ be the projections of the points $M,N,R$ and $S$ on the lines $CD,DA,AB$ and $BC$, respectively. Prove that the points $M,N,R,S,W,X,Y$ and $Z$ lie on a circle.
1989 Czech And Slovak Olympiad IIIA, 1
Three different points $A, B, C $ lying on a circle with center $S$ and a line $p$ perpendicular to $ AS$ are given in the plane. Let's mark the intersections of the line $p$ with the lines $AB$, $AC$ as $D$ and $E$. Prove that the points $B, C, D, E$ lie on the same circle.
2014 Saudi Arabia Pre-TST, 4.4
Let $\vartriangle ABC$ be an acute triangle, with $\angle A> \angle B \ge \angle C$. Let $D, E$ and $F$ be the tangency points between the incircle of triangle and sides $BC, CA, AB$, respectively. Let $J$ be a point on $(BD)$, $K$ a point on $(DC)$, $L$ a point on $(EC)$ and $M$ a point on $(FB)$, such that $$AF = FM = JD = DK = LE = EA.$$Let $P$ be the intersection point between $AJ$ and $KM$ and let $Q$ be the intersection point between $AK$ and $JL$. Prove that $PJKQ$ is cyclic.
2002 All-Russian Olympiad Regional Round, 8.6
Each side of the convex quadrilateral was continued into both sides and on all eight extensions set aside equal segments. It turned out that the resulting $8$ points are the outer ends of the construction the given segments are different and lie on the same circle. Prove that the original quadrilateral is a square.
2006 Switzerland - Final Round, 5
A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.
2012 Switzerland - Final Round, 10
Let $O$ be an inner point of an acute-angled triangle $ABC$. Let $A_1, B_1$ and $C_1$ be the projections of $O$ on the sides $BC, AC$ and $AB$ respectively . Let $P$ be the intersection of the perpendiculars on $B_1C_1$ and $A_1C_1$ from points$ A$ and $B$ respectilvey. Let $H$ be the projection of $P$ on $AB$. Show that points $A_1, B_1, C_1$ and $H$ lie on a circle.
Croatia MO (HMO) - geometry, 2023.7
Given is an acute-angled triangle $ABC$ in which holds $|BC|: |AC| = 3:$2. Let $D$ be the midpoint of the side $\overline{AC}$, and P the midpoint of the segment $\overline{BD}$. A point $X$ is given on the line $AC$ so that $|AX| = |BC|$, where $A$ is between $X$ and $C$. The line $XP$ intersects the side $\overline{BC}$ at point $E$. The line $DE$ intersects the line $AP$ at point $Y$. Prove that the points $A$, $X$, $Y$, $E$ lie on one circle if and only if $|AB| = |BC|$.
2019 Peru EGMO TST, 2
Let $\Gamma$ be the circle of an acute triangle $ABC$ and let $H$ be its orthocenter. The circle $\omega$ with diameter $AH$ cuts $\Gamma$ at point $D$ ($D\ne A$). Let $M$ be the midpoint of the smaller arc $BC$ of $\Gamma$ . Let $N$ be the midpoint of the largest arc $BC$ of the circumcircle of the triangle $BHC$. Prove that there is a circle that passes through the points $D, H, M$ and $N$.
the 13th XMO, P3
Let O be the circumcenter of triangle ABC.
Let H be the orthocenter of triangle ABC.
the perpendicular bisector of AB meet AC,BC at D,E.
the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L.
CH meet FG at T,and ABCT is concyclic.
Prove that LHBC is concyclic.
graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png
2006 Sharygin Geometry Olympiad, 26
Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.
1953 Moscow Mathematical Olympiad, 237
Three circles are pair-wise tangent to each other. Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.
1964 Polish MO Finals, 6
Given is a pyramid $SABCD$ whose base is a convex quadrilateral $ ABCD $ with perpendicular diagonals $ AC $ and $ BD $, and the orthogonal projection of vertex $S$ onto the base is the point $0$ of the intersection of the diagonals of the base. Prove that the orthogonal projections of point $O$ onto the lateral faces of the pyramid lie on the circle.
1960 Poland - Second Round, 5
There are three different points on the line $ A $, $ B $, $ C $ and a point $ S $ outside this line; perpendicularly drawn at points $ A $, $ B $, $ C $ to the lines $ SA $, $ SB $, $ SC $ intersect at points $ M $, $ N $, $ P $. Prove that the points $ M $, $ N $, $ P $, $ S $ lie on the circle
Geometry Mathley 2011-12, 9.3
Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$.
(a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$.
(b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear.
Nguyễn Văn Linh
1993 Tournament Of Towns, (363) 2
Let $O$ be the centre of the circle touching the side $AC$ of triangle $ABC$ and the continuations of the sides $BA$ and $BC$. $D$ is the centre of the circle passing through the points $A$, $B$ and $O$. Prove that the points $A$, $B$, $C$ and $D$ lie on a circle.
(YF Akurlich)
2020 Brazil Cono Sur TST, 4
Let $ABC$ be a triangle and $D$ is a point inside of $\triangle ABC$. The point $A'$ is the midpoint of the arc $BDC$, in the circle which passes by $B,C,D$. Analogously define $B'$ and $C'$, being the midpoints of the arc $ADC$ and $ADB$ respectively. Prove that the four points $D,A',B',C'$ are concyclic.