In trapezoid $ABCD$, the sum of the lengths of the bases $AB$ and $CD$ is equal to the length of the diagonal $BD$. Let $M$ denote the midpoint of $BC$, and let $E$ denote the reflection of $C$ about the line $DM$. Prove that $\angle AEB=\angle ACD$.

Given is a trapezoid $ ABCD$ where $ AB$ and $ CD$ are parallel, and $ A,B,C,D$ are clockwise in this order. Let $ \Gamma_1$ be the circle with center $ A$ passing through $ B$, $ \Gamma_2$ be the circle with center $ C$ passing through $ D$. The intersection of line $ BD$ and $ \Gamma_1$ is $ P$ $ ( \ne B,D)$. Denote by $ \Gamma$ the circle with diameter $ PD$, and let $ \Gamma$ and $ \Gamma_1$ meet at $ X$$ ( \ne P)$. $ \Gamma$ and $ \Gamma_2$ meet at $ Y$. If the circumcircle of triangle $ XBY$ and $ \Gamma_2$ meet at $ Q$, prove that $ B,D,Q$ are collinear.

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic. [i]David Yang.[/i]

Let $O$ be the center of the circumcircle of an acute triangle $ABC$, let $P$ be any point inside the segment $BC$. Suppose the circumcircle of triangle $BPO$ intersects the segment $AB$ at point $R$ and the circumcircle of triangle $COP$ intersects $CA$ at point $Q$. (i) Consider the triangle $PQR$, show that it is similar to triangle $ABC$ and that $O$ is its orthocenter. (ii) Show that the circumcircles of triangles $BPO$, $COP$, $PQR$ have the same radius.

Let $ABC$ be a triangle whose inscribed circle is $\omega$. Let $r_1$ be the line parallel to $BC$ and tangent to $\omega$, with $r_1 \ne BC$ and let $r_2$ be the line parallel to $AB$ and tangent to $\omega$ with $r_2 \ne AB$. Suppose that the intersection point of $r_1$ and $r_2$ lies on the circumscribed circle of triangle $ABC$. Prove that the sidelengths of triangle $ABC$ form an arithmetic progression.

Let $\triangle ABC$ be an acute triangle and let $M, N$ be the midpoints of $AB, AC$ respectively. Given a point $D$ in the interior of segment $BC$ with $DB<DC$, let $P, Q$ the intersections of $DM, DN$ with $AC, AB$ respectively. Let $R \ne A$ be the intersection of circumcircles of triangles $\triangle PAQ$ and $\triangle AMN$. If $K$ is midpoint of $AR$, prove that $\angle MKN=2\angle BAC$

Let $ABC$ be an acute scalene triangle inscribed in a circle \( \Gamma \). The angle bisector of \( \angle BAC \) intersects \( BC \) at \( L \) and \( \Gamma \) at \( S \). The point \( M \) is the midpoint of \( AL \). Let \( AD \) be the altitude in \( \triangle ABC \), and the circumcircle of \( \triangle DSL \) intersects \( \Gamma \) again at \( P \). Let \( N \) be the midpoint of \( BC \), and let \( K \) be the reflection of \( D \) with respect to \( N \). Prove that the triangles \( \triangle MPS \) and \( \triangle ADK \) are similar.

Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.

Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.

Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle). Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.

Given an acute triangle whose circumcenter is $O$.let $K$ be a point on $BC$,different from its midpoint.$D$ is on the extension of segment $AK,BD$ and $AC$,$CD$and$AB$intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

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.

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Given a trapezoid $ABCD$ ($AD \parallel BC$) with $\angle ABC > 90^\circ$ . Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$, respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.

For any $\triangle ABC$, its girth is$l$, its circumradius is$R$, its inscribed radius is $r$.Which one is true? $\text{(A)}l>R+r\qquad\text{(B)}l\leq R+r\qquad\text{(C)}\frac{l}{6}<R+r<6l\qquad\text{(D)}$None above

Let $AB$ be a diameter of a circle $(\omega)$ with centre $O$. From an arbitrary point $M$ on $AB$ such that $MA < MB$ we draw the circles $(\omega_1)$ and $(\omega_2)$ with diameters $AM$ and $BM$ respectively. Let $CD$ be an exterior common tangent of $(\omega_1), (\omega_2)$ such that $C$ belongs to $(\omega_1)$ and $D$ belongs to $(\omega_2)$. The point $E$ is diametrically opposite to $C$ with respect to $(\omega_1)$ and the tangent to $(\omega_1)$ at the point $E$ intersects $(\omega_2)$ at the points $F, G$. If the line of the common chord of the circumcircles of the triangles $CED$ and $CFG$ intersects the circle $(\omega)$ at the points $K, L$ and the circle $(\omega_2)$ at the point $N$ (with $N$ closer to $L$), then prove that $KC = NL$.

Let scalene triangle $ABC$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $BC,CA,$ and $AB$ at $D,E,$ and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let line $DP$ intersect $\omega$ again at $Q \ne D$. The line $OI$ intersects the altitude from $A$ to$ BC$ at $T$. Given that $OI \|BC,$ show that $PQ=PT$.

Let $ABC$ be a triangle with circumcircle $\Gamma$. $D$ is the midpoint of arc $BC$ (this arc does not contain $A$). $E$ is the common point of $BC$ and the perpendicular bisector of $BD$. $F$ is the common point of $AC$ and the parallel to $AB$ containing $D$. $G$ is the common point of $EF$ and $AB$. $H$ is the common point of $GD$ and $AC$. Show that $GAH$ is isosceles.

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

3-There is given a trapezoid $ ABCD$ in the plane with $ BC\parallel{}AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD \equal{} 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP\parallel{}AD$.

Let $ABC$ be an acute-angled triangle, $\Gamma$ its circumcircle and $M$ the midpoint of $BC$. Let $N$ be a point in the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle NAC= \angle BAM$. Let $R$ be the midpoint of $AM$, $S$ the midpoint of $AN$ and $T$ the foot of the altitude through $A$. Prove that $R$, $S$ and $T$ are collinear.

A acute triangle $\triangle ABC$ has circumcenter $O$. The circumcircle of $OAB$, called $O_1$, and the circumcircle of $OAC$, called $O_2$, meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$. Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.

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$.