Found problems: 670
$ABCD$ 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.
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 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 $X$ 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]
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.
The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.
Given a circle with four circles that intersect in pairs as shown in the figure. The "internal" the points of intersection are $A, B, C$ and $D$, while the ‘outer’ points of intersection are $E, F, G$ and $H$. Prove that the quadrilateral $ABCD$ is cyclic if also the quadrilateral $EFGH$ is also cyclic.
[img]https://cdn.artofproblemsolving.com/attachments/0/0/6a369c93e37eefd57775fd8586bdff393e1914.png[/img]
The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.
Let $ABCD$ be a cyclic quadrilateral, so that $|AB| + |CD| = |BC|$. Show that the intersection of the bisector of $\angle DAB$ and $\angle CDA$ lies on the side $BC$.
Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.
On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
Let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$ of a cyclic quadrilateral $ABCD$. Let $P$ be the point of intersection of $AC$ and $BD$. Prove that the circumradii of triangles $PKL, PLM, PMN$ and $PNK$ are equal to one another.
Let $B, C, D, E$ be four pairwise distinct collinear points and let $A$ be a point not on ine $BC$. Now, let the circumcircle of $\triangle ABC$ meet $AD$ and $AE$ respectively again at $F$ and $G$.
Show that $DEFG$ is cyclic if and only if $AB=AC$.
Let $ ABC$ be a triangle and $ AB\ne AC$ . $ D$ is a point on $ BC$ such that $ BA \equal{} BD$ and $ B$ is between $ C$ and $ D$ . Let $ I_{c}$ be center of the circle which touches $ AB$ and the extensions of $ AC$ and $ BC$ . $ CI_{c}$ intersect the circumcircle of $ ABC$ again at $ T$ .
If $ \angle TDI_{c} \equal{} \frac {\angle B \plus{} \angle C}{4}$ then find $ \angle A$
In cyclic quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $E$, while segments $AC$ and $BD$ intersect at $F$. Point $P$ is on ray $EF$ such that angles $BPE$ and $CPE$ are congruent. Prove that angles $APB$ and $DPC$ are also equal.
Let $ABCD$ be quadrilateral inscribed in a circle. Let $M$ be the midpoint of the segment $BD$. If the tangents of the circle at $ B$, and at $D$ are also concurrent with the extension of $AC$, prove that $\angle AMD = \angle CMD$.
Let $ABC$ be a triangle and $M$ be the midpoint of side $BC$. The perpendicular bisector of $BC$ intersects the circumcircle of $\triangle ABC$ at points $K$ and $L$ ($K$ and $A$ lie on the opposite sides of $BC$). A circle passing through $L$ and $M$ intersects $AK$ at points $P$ and $Q$ ($P$ lies on the line segment $AQ$). $LQ$ intersects the circumcircle of $\triangle KMQ$ again at $R$. Prove that $BPCR$ is a cyclic quadrilateral.
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear
Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$.
Trần Quang Hùng
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
[i]Proposed by Anand Iyer[/i]
The median $CM$ is drawn in the triangle $ABC$ intersecting bisector angle $BL$ at point $O$. Ray $AO$ intersects side $BC$ at point $K$, beyond point $K$ draw the segment $KT = KC$. On the ray $BC$ beyond point $C$ draw a segment $CN = BK$. Prove that is a quadrilateral $ABTN$ is cyclic if and only if $AB = AK$.
(Vladislav Yurashev)
Let $ABCD$ be a quadrilateral that can be inscribed in a circle whose diagonals are perpendicular. Denote by $P$ and $Q$ the feet of the perpendiculars through $D$ and $C$ respectively on the line $AB$, $X$ is the intersection point of the lines $AC$ and $DP$, $Y$ is the intersection point of the lines $BD$ and $CQ$. Show that $XY CD$ is a rhombus.
Two sides of the cyclic quadrilateral $ABCD$ are known: $AB = a$, $BC = b$. A point $K$ is taken on the side $CD$ so that $CK = m$. A circle passing through $B$, $K$ and $D$ intersects line $DA$ at a point $M$, different from $D$. Find $AM$.