Found problems: 3585
Let $ABC$ be an acute triangle with $AB < AC < BC$ inscribed in a circle $(c)$ and let $E$ be an arbitrary point on its altitude $CD$. The circle $(c_1)$ with diameter $EC$, intersects the circle $(c)$ at point $K$ (different than $C$), the line $AC$ at point $L$ and the line $BC$ at point $M$. Finally the line $KE$ intersects $AB$ at point $N$. Prove that the quadrilateral $DLMN$ is cyclic.
Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.
[i]Proposed by Le Viet An, Vietnam[/i]
In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. The line through $A$ parallel to $DF$ meets the line through $C$ parallel to $EF$ at $G$.
$(a)$ Prove that the quadrilateral $AICG$ is cyclic.
$(b)$ Prove that the points $B,I,G$ are collinear.
Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $CDX$ and $ABY$ meet line $XY$ again at $P$ and $Q$ respectively. Show that $OP=OQ$.
[i]Proposed by Evan Chang (squareman), USA[/i]
In $\varDelta{ABC}$,the excircle of $A$ is tangent to segment $BC$,line $AB$ and $AC$ at $E,D,F$ respectively.$EZ$ is the diameter of the circle.$B_1$ and $C_1$ are on $DF$, and $BB_1\perp{BC}$,$CC_1\perp{BC}$.Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$,$ZK$ is perpendicular to $FD$ at $K$.If $H$ is the orthocenter of $\varDelta{XYZ}$,prove that:$H,K,X,Y$ are concyclic.
Let $H$ and $O$ be the orthocenter and circumcenter of an acute-angled triangle $ABC$, respectively. The perpendicular bisector of $BH$ meets $AB$ and $BC$ at points $A_1$ and $C_1$, respectively. Prove that $OB$ bisects the angle $A_1OC_1$.
Let $ ABCD$ be a cyclic quadrilateral with $ AB>CD$ and $ BC>AD$. Take points $ X$ and $ Y$ on the sides $ AB$ and $ BC$, respectively, so that $ AX\equal{}CD$ and $ AD\equal{}CY$. Let $ M$ be the midpoint of $ XY$. Prove that $ AMC$ is a right angle.
The internal angle bisectors of $A$, $B$, and $C$ in $\triangle ABC$ concur at $I$ and intersect the circumcircle of $\triangle ABC$ at $L$, $M$, and $N$, respectively. The circle with diameter $IL$ intersects $BC$ at $D$ and $E$; the circle with diameter $IM$ intersects $CA$ at $F$ and $G$; the circle with diameter $IN$ intersects $AB$ at $H$ and $J$. Show that $D$, $E$, $F$, $G$, $H$, and $J$ are concyclic.
Let $ ABCDE$ be a convex pentagon such that
\[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE.
\]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$.
[i]Proposed by Zuming Feng, USA[/i]
Points $M$ and $N$ are taken on the sides $BC$ and $CD$ respectively of a square $ABCD$ so that $\angle MAN=45^{\circ}$. Prove that the circumcentre of $\triangle AMN$ lies on $AC$.
$AB$ is the diameter of circle $O$, $CD$ is a non-diameter chord that is perpendicular to $AB$. Let $E$ be the midpoint of $OC$, connect $AE$ and extend it to meet the circle at point $P$. Let $DP$ and $BC$ meet at $F$. Prove that $F$ is the midpoint of $BC$.
Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.
Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$.
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[*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
[*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$.
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Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$.
Prove that $\angle BXP = \angle CXQ$.
[i]Kian Moshiri, United Kingdom[/i]
Let $BD$ and $CE$ be the altitudes of triangle $ABC$ that intersect at point $H$. Let $F$ be a point on side $AC$ such that $FH\perp CE$. The segment $FE$ intersects the circumcircle of triangle $CDE$ at the point $K$. Prove that $HK\perp EF$ .
(Matthew Kurskyi)
Let $O$ be the circumcenter of the acute triangle $ABC$. Suppose points $A',B'$ and $C'$ are on sides $BC,CA$ and $AB$ such that circumcircles of triangles $AB'C',BC'A'$ and $CA'B'$ pass through $O$. Let $\ell_a$ be the radical axis of the circle with center $B'$ and radius $B'C$ and circle with center $C'$ and radius $C'B$. Define $\ell_b$ and $\ell_c$ similarly. Prove that lines $\ell_a,\ell_b$ and $\ell_c$ form a triangle such that it's orthocenter coincides with orthocenter of triangle $ABC$.
[i]Proposed by Mehdi E'tesami Fard[/i]
Let triangle $\triangle{ABC}$ be acute. Point $D$ is the foot of the altitude of $\triangle{ABC}$ from $A$ to $\overline{BC},$ and $E$ is the foot of the altitude of $\triangle{ABC}$ from $B$ to $\overline{AC}.$ Let $F$ denote the point of intersection between $\overline{BE}$ and $\overline{AD},$ and let $G$ denote the point of intersection between $\overline{CF}$ and $\overline{DE}.$ The areas of triangles $\triangle{EFG}, \triangle{CDG},$ and $\triangle{CEG}$ are $1,4,$ and $3,$ respectively. Find the area of $\triangle{ABC}.$
Let $ ABCD$ be a quadrilateral which is a cyclic quadrilateral and a tangent quadrilateral simultaneously. (By a [i]tangent quadrilateral[/i], we mean a quadrilateral that has an incircle.)
Let the incircle of the quadrilateral $ ABCD$ touch its sides $ AB$, $ BC$, $ CD$, and $ DA$ in the points $ K$, $ L$, $ M$, and $ N$, respectively. The exterior angle bisectors of the angles $ DAB$ and $ ABC$ intersect each other at a point $ K^{\prime}$. The exterior angle bisectors of the angles $ ABC$ and $ BCD$ intersect each other at a point $ L^{\prime}$. The exterior angle bisectors of the angles $ BCD$ and $ CDA$ intersect each other at a point $ M^{\prime}$. The exterior angle bisectors of the angles $ CDA$ and $ DAB$ intersect each other at a point $ N^{\prime}$. Prove that the straight lines $ KK^{\prime}$, $ LL^{\prime}$, $ MM^{\prime}$, and $ NN^{\prime}$ are concurrent.
In triangle $ABC$ let $F$ denote the midpoint of side $BC$. Let the circle passing through point $A$ and tangent to side $BC$ at point $F$ intersect sides $AB$ and $AC$ at points $M$ and $N$, respectively. Let the line segments $CM$ and $BN$ intersect in point $X$. Let $P$ be the second point of intersection of the circumcircles of triangles $BMX$ and $CNX$. Prove that points $A, F$ and $P$ are collinear.
Proposed by Imolay András, Budapest
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.
Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.
[i]Proposed by Greece[/i]
In a triangle $ABC$, $P$ and $Q$ are the feet of the altitudes from $B$ and $A$ respectively. Find the locus of the circumcentre of triangle $PQC$, when point $C$ varies (with $A$ and $B$ fixed) in such a way that $\angle ACB$ is equal to $60^{\circ}$.
Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent to edge in $A$. Let $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.