Found problems: 257
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.
2023 Yasinsky Geometry Olympiad, 4
$ABC$ is an acute triangle and $AD$, $BE$ and $CF$ are the altitudes, with $H$ being the point of intersection of these altitudes. Points $A_1$, $B_1$, $C_1$ are chosen on rays $AD$, $BE$ and $CF$ respectively such that $AA_1 = HD$, $BB_1 = HE$ and $CC_1 =HF$. Let $A_2$, $B_2$ and $C_2$ be midpoints of segments $A_1D$, $B_1E$ and $C_1F$ respectively. Prove that $H$, $A_2$, $B_2$ and $C_2$ are concyclic.
(Mykhailo Barkulov)
1995 Singapore Team Selection Test, 2
Let $ABC$ be an acute-angled triangle. Suppose that the altitude of $\vartriangle ABC$ at $B$ intersects the circle with diameter $AC$ at $P$ and $Q$, and the altitude at $C$ intersects the circle with diameter $AB$ at $M$ and $N$. Prove that $P, Q, M$ and $N$ lie on a circle.
Geometry Mathley 2011-12, 8.3
Let $ABC$ be a scalene triangle, $(O)$ and $H$ be the circumcircle and its orthocenter. A line through $A$ is parallel to $OH$ meets $(O)$ at $K$. A line through $K$ is parallel to $AH$, intersecting $(O)$ again at $L$. A line through $L$ parallel to $OA$ meets $OH$ at $E$. Prove that $B,C,O,E$ are on the same circle.
Trần Quang Hùng
2022 Bulgarian Autumn Math Competition, Problem 9.2
Given is the triangle $ABC$ such that $BC=13, CA=14, AB=15$ Prove that $B$, the incenter $J$ and the midpoints of $AB$ and $BC$ all lie on a circle
2006 Korea Junior Math Olympiad, 3
In a circle $O$, there are six points, $A,B,C,D,E, F$ in a counterclockwise order. $BD \perp CF$, and $CF,BE,AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE // MN$.
VI Soros Olympiad 1999 - 2000 (Russia), 11.4
Given isosceles triangle $ABC$ ($AB = AC$). A straight line $\ell$ is drawn through its vertex $B$ at a right angle with $AB$ . On the straight line $AC$, an arbitrary point $D$ is taken, different from the vertices, and a straight line is drawn through it at a right angle with $AC$, intersecting $\ell$ at the point $F$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the circumscribed circle of triangle $ABD$.
2019 Polish MO Finals, 1
Let $ABC$ be an acute triangle. Points $X$ and $Y$ lie on the segments $AB$ and $AC$, respectively, such that $AX=AY$ and the segment $XY$ passes through the orthocenter of the triangle $ABC$. Lines tangent to the circumcircle of the triangle $AXY$ at points $X$ and $Y$ intersect at point $P$. Prove that points $A, B, C, P$ are concyclic.
2021 Sharygin Geometry Olympiad, 10-11.3
The bisector of angle $A$ of triangle $ABC$ ($AB > AC$) meets its circumcircle at point $P$. The perpendicular to $AC$ from $C$ meets the bisector of angle $A$ at point $K$. A cừcle with center $P$ and radius $PK$ meets the minor arc $PA$ of the circumcircle at point $D$. Prove that the quadrilateral $ABDC$ is circumscribed.
Mexican Quarantine Mathematical Olympiad, #3
Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic.
[i]Proposed by Ariel García[/i]
2021 Chile National Olympiad, 4
Consider quadrilateral $ABCD$ with $|DC| > |AD|$. Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$. Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $ \angle ABC$. Let us call $X = L_1 \cap L_2$. Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$
[img]https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png[/img]
2019 Brazil EGMO TST, 3
Let $ABC$ be a triangle and $E$ and $F$ two arbitrary points on sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. The point $D$ is such that $EF$ bisects the segment $MD$ . Finally, $O$ is the circumcenter of triangle $ABC$. Prove that $D$ lies on line $BC$ if and only if $O$ lies on the circumcircle of triangle $AEF$.
2020 Regional Competition For Advanced Students, 3
Let a triangle $ABC$ be given with $AB <AC$. Let the inscribed center of the triangle be $I$. The perpendicular bisector of side $BC$ intersects the angle bisector of $BAC$ at point $S$ and the angle bisector of $CBA$ at point $T$. Prove that the points $C, I, S$ and $T$ lie on a circle.
(Karl Czakler)
Croatia MO (HMO) - geometry, 2016.3
Given a cyclic quadrilateral $ABCD$ such that the tangents at points $B$ and $D$ to its circumcircle $k$ intersect at the line $AC$. The points $E$ and $F$ lie on the circle $k$ so that the lines $AC, DE$ and $BF$ parallel. Let $M$ be the intersection of the lines $BE$ and $DF$. If $P, Q$ and $R$ are the feet of the altitides of the triangle $ABC$, prove that the points $P, Q, R$ and $M$ lie on the same circle
2020 Dutch BxMO TST, 2
In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.
2001 Kazakhstan National Olympiad, 7
Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.
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 $ABC$.
2013 Estonia Team Selection Test, 4
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
2021 China Girls Math Olympiad, 7
In an acute triangle $ABC$, $AB \neq AC$, $O$ is its circumcenter. $K$ is the reflection of $B$ over $AC$ and $L$ is the reflection of $C$ over $AB$. $X$ is a point within $ABC$ such that $AX \perp BC, XK=XL$. Points $Y, Z$ are on $\overline{BK}, \overline{CL}$ respectively, satisfying $XY \perp CK, XZ \perp BL$.
Proof that $B, C, Y, O, Z$ lie on a circle.
2017 NZMOC Camp Selection Problems, 6
Let $ABCD$ be a quadrilateral. The circumcircle of the triangle $ABC$ intersects the sides $CD$ and $DA$ in the points $P$ and $Q$ respectively, while the circumcircle of $CDA$ intersects the sides $AB$ and $BC$ in the points $R$ and $S$. The lines $BP$ and $BQ$ intersect the line $RS$ in the points $M$ and $N$ respectively. Prove that the points $M, N, P$ and $Q$ lie on the same circle.
Estonia Open Junior - geometry, 2000.2.4
In the plane, there is an acute angle $\angle AOB$ . Inside the angle points $C$ and $D$ are chosen so that $\angle AOC = \angle DOB$. From point $D$ the perpendicular on $OA$ intersects the ray $OC$ at point $G$ and from point C the perpendicular on $OB$ intersects the ray $OD$ at point $H$. Prove that the points $C, D, G$ and $H$ are conlyclic.
2020 Dutch BxMO TST, 2
In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.
Estonia Open Junior - geometry, 2004.2.3
Circles $c_1$ and $c_2$ with centres $O_1$and $O_2$, respectively, intersect at points $A$ and $B$ so that the centre of each circle lies outside the other circle. Line $O_1A$ intersects circle $c_2$ again at point $P_2$ and line $O_2A$ intersects circle $c_1$ again at point $P_1$. Prove that the points $O_1,O_2, P_1, P_2$ and $B$ are concyclic
Estonia Open Senior - geometry, 2008.1.2
Let $O$ be the circumcentre of triangle $ABC$. Lines $AO$ and $BC$ intersect at point $D$. Let $S$ be a point on line $BO$ such that $DS \parallel AB$ and lines $AS$ and $BC$ intersect at point $T$. Prove that if $O, D, S$ and $T$ lie on the same circle, then $ABC$ is an isosceles triangle.
2017 Ukrainian Geometry Olympiad, 4
In the right triangle $ABC$ with hypotenuse $AB$, the incircle touches $BC$ and $AC$ at points ${{A}_{1}}$ and ${{B}_{1}}$ respectively. The straight line containing the midline of $\Delta ABC$ , parallel to $AB$, intersects its circumcircle at points $P$ and $T$. Prove that points $P,T,{{A}_{1}}$ and ${{B}_{1}}$ lie on one circle.