Found problems: 311
The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$
Let $ABC$ be a triangle with $AC = 28$, $BC = 33$, and $\angle ABC = 2\angle ACB$. Compute the length of side $AB$.
[i]2016 CCA Math Bonanza #10[/i]
Triangle $\vartriangle ABC$ has $AB = 8$, $AC = 10$, and $AD =\sqrt{33}$, where $D$ is the midpoint of $BC$. Perpendiculars are drawn from $D$ to meet $AB$ and $AC$ at $E$ and $F$, respectively. The length of $EF$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a, c$ are relatively prime and $b$ is square-free. Compute $a + b + c$.
Let $O$ be the center of the equilateral triangle $ABC$. Pick two points $P_1$ and $P_2$ other than $B$, $O$, $C$ on the circle $\odot(BOC)$ so that on this circle $B$, $P_1$, $P_2$, $O$, $C$ are placed in this order. Extensions of $BP_1$ and $CP_1$ intersects respectively with side $CA$ and $AB$ at points $R$ and $S$. Line $AP_1$ and $RS$ intersects at point $Q_1$. Analogously point $Q_2$ is defined. Let $\odot(OP_1Q_1)$ and $\odot(OP_2Q_2)$ meet again at point $U$ other than $O$.
Prove that $2\,\angle Q_2UQ_1 + \angle Q_2OQ_1 = 360^\circ$.
Remark. $\odot(XYZ)$ denotes the circumcircle of triangle $XYZ$.
Let a point $P$ inside a parallelogram $ABCD$ be given such that $\angle APB +\angle CPD = 180^o$. Prove that $AB \cdot AD = BP \cdot DP + AP \cdot CP$.
Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to $\sqrt{2}$.
In $ \triangle ABC$ we have $ AB \equal{} 7$, $ AC \equal{} 8$, and $ BC \equal{} 9$. Point $ D$ is on the circumscribed circle of the triangle so that $ \overline{AD}$ bisects $ \angle BAC$. What is the value of $ AD/CD$?
$ \textbf{(A)}\ \frac{9}{8}\qquad
\textbf{(B)}\ \frac{5}{3}\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ \frac{17}{7}\qquad
\textbf{(E)}\ \frac{5}{2}$
Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$ ; let $G$ be the centroid of $\Delta{ABC}$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta{ABC}$.
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.