Found problems: 670
Let $ABCD$ be a cyclic quadrilateral satysfing $AB=AD$ and $AB+BC=CD$. Determine $\measuredangle CDA$.
Let $ABCD$ be a convex cyclic quadrilateral with $AD > BC$, A$B$ not being diameter and $C D$ belonging to the smallest arc $AB$ of the circumcircle. The rays $AD$ and $BC$ are cut at $K$, the diagonals $AC$ and $BD$ are cut at $P$ and the line $KP$ cuts the side $AB$ at point $L$. Prove that angle $\angle ALK$ is acute.
Let $ABCD$ be a cyclie quadrilateral, $\omega$ be it's circumcircle and $M$ be the midpoint of the arc $AB$ of $\omega$ which does not contain the vertices $C$ and $D$. The line that passes through $M$ and the intersection point of segments $AC$ and $BD$, intersects again $\omega$ in $N$. Let $P$ and $Q$ be points in the $CD$ segment such that $\angle AQD = \angle DAP$ and $\angle BPC = \angle CBQ$. Prove that the circumcircle of $NPQ$ and $\omega$ are tangent to each other.
In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.
Quadrilateral $ABCD$ is a cyclic, $AB = AD$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ respectfully so that $\angle MAN =1/2 (\angle BAD)$. Prove that $MN = BM + ND$.
[i](5 points)[/i]
A quadrilateral $ABCD$ without parallel sides is inscribed in a circle $\omega$. We draw a line $\ell_a \parallel BC$ through the point $A$, a line $\ell_b \parallel CD$ through the point $B$, a line $\ell_c \parallel DA$ through the point $C$, and a line $\ell_d \parallel AB$ through the point $D$. Suppose that the quadrilateral whose successive sides lie on these four straight lines is inscribed in a circle $\gamma$ and that $\omega$ and $\gamma$ intersect in points $E$ and $F$. Show that the lines $AC, BD$ and $EF$ intersect in one point.
[i]Proposed by A. Kuznetsov[/i]
$ABCD$ is a cyclic quadrilateral in which $AB=4$, $BC=3$, $CD=2$, and $AD=5$. Diagonals $AC$ and $BD$ intersect at $X$. A circle $\omega$ passes through $A$ and is tangent to $BD$ at $X$. $\omega$ intersects $AB$ and $AD$ at $Y$ and $Z$ respectively. Compute $YZ/BD$.
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.
[i]Robin Son[/i]
Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively.
a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$.
b) Does the converse implication also always hold?
Let $ ABCD$ be a cyclic quadrilateral and let $ E$ and $ F$ be the feet of perpendiculars from the intersection of diagonals $ AC$ and $ BD$ to $ AB$ and $ CD$, respectively. Prove that $ EF$ is perpendicular to the line through the midpoints of $ AD$ and $ BC$.
Let $ABC$ be an acute triangle and $M$ be the midpoint of $AB$. A circle through the points $B$ and $C$ intersects the segments $CM$ and $BM$ at points $P$ and $Q$ respectively. Point $K$ is symmetric to $P$ with respect to point $M$. The circumcircles of $\triangle AKM$ and $\triangle CQM$ intersect for the second time at $X$. The circumcircles of $\triangle AMC$ and $\triangle KMQ$ intersect for the second time at $Y$. The segments $BP$ and $CQ$ intersect at point $T$. Prove that the line $MT$ is tangent to the circumcircle of $\triangle MXY$.
Given a cyclic $ABCD$ with $AB=AD$. Points $M$ and $N$ are marked on the sides $CD$ and $BC$, respectively, so that $DM+BN=MN$. Prove that the circumcenter of the triangle $AMN$ belongs to the segment $AC$.
N.Sedrakian
$ABCD$ is a cyclic quadrilateral and $M$ a point on the side $CD$ such that $ADM$ and $ABCM$ have the same area and the same perimeter. Show that two sides of $ABCD$ have the same length.
Let $ABCD$ be a cyclic quadrilateral with opposite sides not parallel. Let $X$ and $Y$ be the intersections of $AB,CD$ and $AD,BC$ respectively. Let the angle bisector of $\angle AXD$ intersect $AD,BC$ at $E,F$ respectively, and let the angle bisectors of $\angle AYB$ intersect $AB,CD$ at $G,H$ respectively. Prove that $EFGH$ is a parallelogram.
Let $ABC$ be a triangle. $M$ the midpoint of $AB$ and $L$ the midpoint of $BC$. We denote by $G$ the intersection of $AL$ with $CM$ and we take $E$ a point such that $G$ is the midpoint of the segment $AE$. Prove that the quadrilateral $MCEB$ is cyclic if and only if $MB = BG$.
Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.
An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.
The diagonals of the cyclic quadrilateral $ABCD$ intersect at the point $E$. Let $P$ and $Q$ are the centers of the circles circumscribed around the triangles $BCE$ and $DCE$, respectively. A straight line passing through the point $P$ parallel to $AB$, and a straight line passing through the point $Q$ parallel to $AD$, intersect at the point $R$. Prove that the point $R$ lies on segment $AC$.
A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.
Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?
Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.
Find the angles of the cyclic quadrilateral if you know that each of its diagonals is a bisector of one angle and a trisector of the opposite one (the trisector of the angle is one of the two rays that lie in the interior of the angle and divide it into three equal parts).
(Vyacheslav Yasinsky)
Prove that in a convex quadrilateral inscribed in a circle, straight lines passing through the midpoints of the sides and perpendicular to the opposite sides intersect at one point.