Found problems: 3585
Let $\triangle ABC$ be an acute triangle with orthocenter $H$. The circle $\Gamma$ with center $H$ and radius $AH$ meets the lines $AB$ and $AC$ at the points $E$ and $F$ respectively. Let $E'$, $F'$ and $H'$ be the reflections of the points $E$, $F$ and $H$ with respect to the line $BC$, respectively. Prove that the points $A$, $E'$, $F'$ and $H'$ lie on a circle.
[i]Proposed by Jasna Ilieva[/i]
Let $O$, $I$, and $\omega$ be the circumcenter, the incenter, and the incircle of nonequilateral $\triangle ABC$. Let $\omega_A$ be the unique circle tangent to $AB$ and $AC$, such that the common chord of $\omega_A$ and $\omega$ passes through the center of $\omega_A$ . Let $O_A$ be the center of $\omega_A$. Define $\omega_B, O_B, \omega_C, O_C$ similarly. If $\omega$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively, prove that the perpendiculars from $D$, $E$, $F$ to $O_BO_C , O_CO_A , O_AO_B$ are concurrent on the line $OI$.
[i]Pitchayut Saengrungkongka[/i]
$ABCD$ is a cyclic quadrilateral in which $AB=3$, $BC=5$, $CD=6$, and $AD=10$. $M$, $I$, and $T$ are the feet of the perpendiculars from $D$ to lines $AB$, $AC$, and $BC$ respectively. Determine the value of $MI/IT$.
A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$, and the diagonals of $ABCD$ meet in point $Y$. Prove that points $X, Y, I$ and $M$ are collinear.
Let $ABC$ be a scalene triangle, with circumcircle $\omega$ and incentre $I.{}$ The tangent line at $C$ to $\omega$ intersects the line $AB$ at $D.{}$ The angle bisector of $BDC$ meets $BI$ at $P{}$ and $AI{}$ at $Q{}.$ Let $M{}$ be the midpoint of the segment $PQ.$ Prove that the line $IM$ passes through the middle of the arc $ACB$ of $\omega.$
[i]Dana Heuberger[/i]
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$.
Show that $A,X,Y$ are collinear.
Let $ABC$ be a triangle. Choose a point $D$ in its interior. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ be a circle passing through $C$ and $D$ so that the other point of intersection of the two circles lies on $AD$. Let $\omega_1$ and $\omega_2$ intersect side $BC$ at $E$ and $F$, respectively. Denote by $X$ the intersection of $DF$, $AB$ and $Y$ the intersection of $DE, AC$. Show that $XY \parallel BC$.
A circle $k$ with center $O$ and radius $r$ and a line $p$ which has no common points with $k$, are given. Let $E$ be the foot of the perpendicular from $O$ to $p$. Let $M$ be an arbitrary point on $p$, distinct from $E$. The tangents from the point $M$ to the circle $k$ are $MA$ and $MB$. If $H$ is the intersection of $AB$ and $OE$, then prove that $OH=\frac{r^2}{OE}$.
In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.
Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$.
Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point.
[i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]
In cyclic quadrilateral $ABCD$, $\angle A+ \angle D=\frac{\pi}{2}$. $AC$ intersects $BD$ at ${E}$. A line ${l}$ cuts segment $AB, CD, AE, DE$ at $X, Y, Z, T$ respectively. If $AZ=CE$ and $BE=DT$, prove that the diameter of the circumcircle of $\triangle EZT$ equals $XY$.
Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.
In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$.
Proposed by Fatemeh Sajadi
Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $D$ be the midpoint of side $BC$ and $BE,CZ$ be the altitudes of the triangle $ABC$. Line $ZE$ intersects line $BC$ at point $O$.
(i) Find all the angles of the triangle $ZDE$ in terms of angle $\angle A$ of the triangle $ABC$
(ii) Find the angle $\angle BOZ$ in terms of angles $\angle B$ and $\angle C$ of the triangle $ABC$
Let $ABCD$ be a parallelogram in which angle at $B$ is obtuse and $AD>AB$. Points $K$ and $L$ on $AC$ such that $\angle ADL=\angle KBA$(the points $A, K, C, L$ are all different, with $K$ between $A$ and $L$). The line $BK$ intersects the circumcircle $\omega$ of $ABC$ at points $B$ and $E$, and the line $EL$ intersects $\omega$ at points $E$ and $F$. Prove that $BF||AC$.
Let $X$ be a point inside convex quadrilateral $ABCD$ with $\angle AXB+\angle CXD=180^{\circ}$. If $AX=14$, $BX=11$, $CX=5$, $DX=10$, and $AB=CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$.
[i]Proposed by Michael Kural[/i]
Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
[i]Proposed by Jack Edward Smith, UK[/i]
Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$
Given a triangle $ABC$ with $AB=AC$ and circumcenter $O$. Let $D$ and $E$ be midpoints of $AC$ and $AB$ respectively, and let $DE$ intersect $AO$ at $F$. Denote $\omega$ to be the circle $(BOE)$. Let $BD$ intersect $\omega$ again at $X$ and let $AX$ intersect $\omega$ again at $Y$.
Suppose the line parallel to $AB$ passing through $O$ meets $CY$ at $Z$. Prove that the lines $FX$ and $BZ$ meet at $\omega$.
[i]Proposed by Ivan Chan Kai Chin[/i]
Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$.
Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.
Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $I$ be the incentre of triangle $ABC$, and let $D,E,F$ be the points where the incircle touches the sides $BC,CA,AB,$ respectively. Let $BI,CI$ meet the line $EF$ at $Y,X$ respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that
$\text{(i)}$ $B,C,Y,X$ are concyclic.
$\text{(ii)}$ $I$ is also the incentre of triangle $DYX$.
Points $A, B$ and $P$ lie on the circumference of a circle $\Omega_1$ such that $\angle APB$ is an obtuse angle. Let $Q$ be the foot of the perpendicular from $P$ on $AB$. A second circle $\Omega_2$ is drawn with centre $P$ and radius $PQ$. The tangents from $A$ and $B$ to $\Omega_2$ intersect $\Omega_1$ at $F$ and $H$ respectively. Prove that $FH$ is tangent to $\Omega_2$.
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.
Prove that the lines $AP$ and $CS$ are parallel.
The diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$ intersect at a point O. It is known that $\angle BAD = 60^o$ and $AO = 3OC$. Prove that the sum of some two sides of a quadrilateral is equal to the sum of the other two sides.
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.