Found problems: 3585
Scalene triangle $ABC$ satisfies $\angle A = 60^{\circ}$. Let the circumcenter of $ABC$ be $O$, the orthocenter be $H$, and the incenter be $I$. Let $D$, $T$ be the points where line $BC$ intersects the internal and external angle bisectors of $\angle A$, respectively. Choose point $X$ on the circumcircle of $\triangle IHO$ such that $HX \parallel AI$. Prove that $OD \perp TX$.
Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$.
Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.
The diagonals of a quadrangle inscribed in a circle intersect at point $K$. The projections of the point $ K$ onto the subsequent sides of this quadrangle are points $M, N, P, Q$. Prove that these lines $KM$, $KN$, $KP$, $KQ$ are the angle bisectors of the quadrangle $MNPQ$.
Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$. A variable point $X$ is chosen on minor arc $AB$ of $\omega$, and segments $CX$ and $AB$ meet at $D$. Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$, respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized.
[i]Proposed by Zuming Feng[/i]
In acute triangle $ABC$ ($AB<AC$) point $O$ is center of its circumcircle $\Omega$. Let the tangent to $\Omega$ drawn at point $A$ intersect the line $BC$ at point $D$. Let the line $DO$ intersects the segments $AB$ and $AC$ at points $E$ and $F$, respectively. Point $G$ is constructed such that $AEGF$ is a parallelogram. Let $K$ and $H$ be points of intersection of segment $BC$ with segments $EG$ and $FG$, respectively. Prove that the circle $(GKH)$ touches the circle $\Omega$.
[i] Proposed by Dong Luu [/i]
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 $TB = TQ$.
Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.
In a cyclic quadrilateral $ABCD$, $AB=a$, $BC=b$, $CD=c$, $\angle ABC = 120^\circ$ and $\angle ABD = 30^\circ$. Prove that
(1) $c \ge a + b$;
(2) $|\sqrt{c + a} - \sqrt{c + b} | = \sqrt{c - a - b}$.
[size=130]In $\triangle ABC$, $\angle A=60^\circ$. $\odot I$ is the incircle of $\triangle ABC$. $\odot I$ is tangent to sides $AB$, $AC$ at $D$, $E$, respectively. Line $DE$ intersects line $BI$ and $CI$ at $F$, $G$ respectively. Prove that [/size]$FG=\frac{BC}{2}$.
A point $P$ lies inside a convex quadrilateral $ABCD$. Common internal tangents to the incircles of triangles $PAB$ and $PCD$ meet at point $Q$, and common internal tangents to the incircles of $PBC,PAD$ meet at point $R$. Prove that $P,Q,R$ are collinear.
let $ABCD$ be a cyclic quadrilateral with $E$,an interior point such that $AB=AD=AE=BC$. Let $DE$ meet the circumcircle of $BEC$ again at $F$. Suppose a common tangent to the circumcircle of $BEC$ and $DEC$ touch the circles at $F$ and $G$ respectively. Show that $GE$ is the external angle bisector of angle $BEF$
Let $ABC$ be an acute triangle and $D$ a point on the side $AB$. The circumcircle of triangle $BCD$ cuts the side $AC$ again at $E$ .The circumcircle of triangle $ACD$ cuts the side $BC$ again at $F$. If $O$ is the circumcenter of the triangle $CEF$. Prove that $OD$ is perpendicular to $AB$.
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
A pentagon is inscribed in a circle, such that the pentagon has an incircle. All $10$ sets of $3$ vertices from the pentagon are chosen, and the incenters of each of the $10$ resulting triangles are drawn in. Prove these $10$ incenters lie on $2$ concentric circles.
Note: I spent nearly no time on this, so if anyone took CMO and I misremembered just let me know.
Let the intersection of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ be point $E$. Let $M$ be the midpoint of $AE$ and $N$ be the midpoint of $CD$. It's known that $BD$ bisects $\angle ABC$. Prove that $ABCD$ is cyclic if and only if $MBCN$ is cyclic.
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.
[i]Proposed by David Stoner[/i]
In a triangle $ABC$ with $AB\neq AC$, the internal and external bisectors of angle $A$ meet the line $BC$ at $D$ and $E$ respectively. If the feet of the perpendiculars from a point $F$ on the circle with diameter $DE$ to $BC,CA,AB$ are $K,L,M$, respectively, show that $KL=KM$.
Let $A$, $B$, $C$, $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$. Let $P$, $Q$, $R$ be the orthocenters of $\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively. Suppose that the lines $AP$, $BQ$, $CR$ are pairwise distinct and are concurrent. Show that the four points $A$, $B$, $C$, $D$ lie on a circle.
[i]Andrew Gu[/i]
The incircle of triangle $ABC$ touches sides $AB$;$BC$;$CA$ at $M$;$N$;$K$, respectively.
The line through $A$ parallel to $NK$ meets $MN$ at $D$.
The line through $A$ parallel to $MN$ meets $NK$ at $E$.
Show that the line $DE$ bisects sides $AB$ and $AC$ of triangle $ABC$.
[i]M. Sonkin[/i]
(A.Zaslavsky) The circumradius of triangle $ ABC$ is equal to $ R$. Another circle with the same radius passes through the orthocenter $ H$ of this triangle and intersect its circumcirle in points $ X$, $ Y$. Point $ Z$ is the fourth vertex of parallelogram $ CXZY$. Find the circumradius of triangle $ ABZ$.
$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$
Let ABCD be a convex quadrilateral, and let M and N be the midpoints of its sides AD and BC, respectively. Assume that the points A, B, M, N are concyclic, and the circumcircle of triangle BMC touches the line AB. Show that the circumcircle of triangle AND touches the line AB, too.
Darij
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.
Let ${ABC}$ be an acute angled triangle, let ${O}$ be its circumcentre, and let ${D,E,F}$ be points on the sides ${BC,CA,AB}$, respectively. The circle ${(c_1)}$ of radius ${FA}$, centered at ${F}$, crosses the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centered at $D$, crosses the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$. Finally, the circle ${(c_3)}$ of radius $EC$, centered at $E$, crosses the segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$. Prove that the quadrilaterals $BKF{A}',CLD{B}'$ and $AME{C}'$ are all cyclic, and their circumcircles share a common point.
Evangelos Psychas (Greece)
The quadrilateral ABCD is inscribed in a circle which has diameter BD. Points A’ and B’ are symmetric to A and B with respect to the line BD and AC respectively. If the lines A’C, BD intersect at P and AC, B’D intersect at Q, prove that PQ is perpendicular to AC.