Found problems: 670
Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.
In a cyclic quadrilateral $ABCD$ with circumcenter $O$, the diagonals $AC$ and $BD$ intersect at $X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect at $Y$. Let the circumcircles of triangles $AXB$ and $CXD$ intersect at $Z$. If $O$ lies inside $ABCD$ and if the points $O,X,Y,Z$ are all distinct, prove that $O,X,Y,Z$ lie on a circle.
Let $K, L, M$ denote three points on the sides $BC$, $AB$ and $BC$ of $\triangle{ABC}$, so that $ALKM$ is a parallelogram. Points $S$ and $T$ are chosen on lines $KL$ and $KM$ respectively, so that the quadrilaterals $AKBS$ and $AKCT$ are both cyclic. Prove that $MLST$ is cyclic if and only if $K$ is the midpoint of $BC$.
Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$.
Let $ABCD$ be a cyclic quadrilateral. Let $ P$ be the intersection of the lines $BC$ and $AD$. Line $AC$ cuts the circumscribed circle of the triangle $BDP$ in $S$ and $T$, with $S$ between $ A$ and $C$. The line $BD$ intersects the circumscribed circle of the triangle $ACP$ in $U$ and $V$, with $U$ between $ B$ and $D$. Prove that $PS = PT = PU = PV$.
Let $\triangle ABC$ be an acute triangle with circumcenter $O$, and let $Q\neq A$ denote the point on $\odot (ABC)$ for which $AQ\perp BC$. The circumcircle of $\triangle BOC$ intersects lines $AC$ and $AB$ for the second time at $D$ and $E$ respectively. Suppose that $AQ$, $BC$, and $DE$ are concurrent. If $OD=3$ and $OE=7$, compute $AQ$.
Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \ne A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T$, $E$, $B$, $Z$ are concyclic.
Proposed by [i]Theoklitos Parayiou, Cyprus[/i]
Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively.
Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.
A quadrilateral $ABCD$ is inscribed in a circle and its two diagonals $AC,BD$ meet at $G$. Let $M$ be the center of $CD, E,F$ be the points on $BC, AD$ respectively such that $ME \parallel AC$ and $MF \parallel BD$. Point $H$ is the projection of $G$ onto $CD$. The circumcircle of $MEF$ meets $CD$ at $N$ distinct from $M$. Prove that $MN = MH$
Tran Quang Hung, Nguyen Le Phuoc, Thanh Xuan, Hanoi
Let $ABCD$ be a cyclic quadrilateral with $AB<CD$. The diagonals intersect at the point $F$ and lines $AD$ and $BC$ intersect at the point $E$. Let $K$ and $L$ be the orthogonal projections of $F$ onto lines $AD$ and $BC$ respectively, and let $M$, $S$ and $T$ be the midpoints of $EF$, $CF$ and $DF$ respectively. Prove that the second intersection point of the circumcircles of triangles $MKT$ and $MLS$ lies on the segment $CD$.
[i](Greece - Silouanos Brazitikos)[/i]
The quadrilateral $ABCD$ can be inscribed in a circle and $\angle{ABD}$ is a right angle. $M$ is the midpoint of $BD$, where $CM$ is an altitude of $\triangle{BCD}$. If $AB=14$ and $CD=6\sqrt{11}$, what [is] the length of $AD$?
$\text{(A) }36\qquad\text{(B) }38\qquad\text{(C) }41\qquad\text{(D) }42\qquad\text{(E) }44$
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
On the sides $AB$ and $BC$ of triangle $ABC$, points $K$ and $M$ are chosen such that the quadrilaterals $AKMC$ and $KBMN$ are cyclic , where $N = AM \cap CK$ . If these quads have the same circumradii, find $\angle ABC$
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .
Given is a cyclic quadrilateral $ABCD$ with $|AB| = |BC|$. Point $E$ is on the arc $CD$ where $A$ and $B$ are not on. Let $P$ be the intersection point of $BE$ and $CD$ , let $Q$ be the intersection point of $AE$ and $BD$ . Prove that $PQ \parallel AC$.
The external bisectors of the angles of the convex quadrilateral $ABCD$ intersect each other in $E,F,G$ and $H$ such that $A\in EH, \ B\in EF, \ C\in FG, \ D\in GH$. We know that the perpendiculars from $E$ to $AB$, from $F$ to $BC$ and from $G$ to $CD$ are concurrent. Prove that $ABCD$ is cyclic.
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively.
(a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic.
(b) If $R$ is the radius of $\Omega$ and $d$ is the distance between its centre and $P$, then find the radius of the circle in (a) in terms of $R$ and $d$.
A quadrangle was drawn on the board, that you can inscribe and circumscribe a circle. Marked are the centers of these circles and the intersection point of the lines connecting the midpoints of the opposite sides, after which the quadrangle itself was erased. Restore it with a compass and ruler.
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]
Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to
$ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.
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 $\vartriangle ABC$.
If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.
Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.
Let $W,X,Y$ and $Z$ be points on a circumference $\omega$ with center $O$, in that order, such that $WY$ is perpendicular to $XZ$; $T$ is their intersection. $ABCD$ is the convex quadrilateral such that $W,X,Y$ and $Z$ are the tangency points of $\omega$ with segments $AB,BC,CD$ and $DA$ respectively. The perpendicular lines to $OA$ and $OB$ through $A$ and $B$, respectively, intersect at $P$; the perpendicular lines to $OB$ and $OC$ through $B$ and $C$, respectively, intersect at $Q$, and the perpendicular lines to $OC$ and $OD$ through $C$ and $D$, respectively, intersect at $R$. $O_1$ is the circumcenter of triangle $PQR$. Prove that $T,O$ and $O_1$ are collinear.
[i]Proposed by CDMX[/i]