Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pariwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum_{i=1}^\infty C_{n-2}C_{n-1} = 6p$. Find $p$.
Squares $ABCD$ and $EFGH$ have a common center at $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$.
Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP < AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
As shown, in quadrilateral $ABCD$, $AB=AD$, $\angle{BAD} = \angle{DCB} = 90^\circ$. Draw altitude from $A$ towards $BC$ and let the foot be $E$. If $AE=1$, find the area of $ABCD$.
In pentagon $ABCDE$, if $AB=AE$, $BC+DE=CD$, and $\angle{ABC} + \angle{AED} = 180^\circ$, show that $\angle{ADE}=\angle{ADC}$.
Let point $P$ be inside equilateral $\triangle{ABC}$ such that $PA=2$, $PB=2\sqrt{3}$, and $PC=4$. Find the side length of this equilateral triangle.
Let point $P$ be inside of equilateral $\triangle{ABC}$ such that $PA=3, PB=4,$ and $PC=5$. Find the measurement of $\angle{APB}$.
Let point $P$ be inside equilateral $\triangle{ABC}$ such that $\angle{APB}=115^\circ$ and $\angle{BPC}=125^\circ$. Find the measurements of the three internal angles if we construct a triangle using $PA$, $PB$, and $PC$.
Let $\triangle{ABC}$ be an isosceles right triangle where $\angle{C}=90^\circ$. If points $M$ and $N$ are on $AB$ such that $\angle{MCN}=45^\circ$, $AM=4$, and $BN=3$, find the length of $MN$.
Given an acute $\triangle$, let $D$ be the middle point of $AB$, and $DE\perp DF$ where points $E$ and $F$ are on the other two sides respectively. Show that $S_{\triangle{DEF}} < S_{\triangle{ADF}} + S_{\triangle{BDE}}$
In $\triangle{ABC}$, let $AB=c$, $AC=b$, and $\angle{BAC}=\alpha$. If $AD$ bisects $\angle{BAC}$ and intersects $BC$ at $D$, find the length of $AD$.
As shown, prove $$\frac{\sin(\alpha+\beta)}{PC}=\frac{\sin{\alpha}}{PB}+\frac{\sin{\beta}}{PA}$$
(Weitzenbock's Inequality) Let $a, b, c$, and $S$ be a triangle's three sides' lengths and its area, respectively. Show that $$a^2 + b^2 + c^2 \ge 4\sqrt{3}\cdot S$$
As show, three squares are arranged side-by-side such that their bases are collinear. The sides of two squares are known and marked. Find the area of shaded triangle.
Let $ABCD$ be a rectangle where $AB=4$ and $BC=6$. If $AE=CG=3$, $BF=DH=4$, and $S_{AEPH}=5$. Find the area of $PFCG$.
Let $ABCD$ be a rectangle where $AB=3$ and $AD=4$. Point $P$ is on the side $AD$. If points $E$ and $F$ are on $AC$ and $BD$ respectively such that $PE \perp AC$ and $PF \perp BD$. Compute $PE+PF$.
As shown in the diagram, both $ABCD$ and $BEFG$ are squares, where point $E$ is on $AB$. If $AD=2$, compute the area of $\triangle{AFC}$.
Let $P$ be a point inside $\triangle{ABC}$. If $AP$, $BP$, and $CP$ intersect the opposite sides at $D$, $E$, and $F$, respectively. Show that $$\frac{PD}{AD}+\frac{PE}{BE}+\frac{PF}{CF}=1$$
Let real numbers $x_1$ and $x_2$ satisfy $ \frac{\pi}{2} > x_1 > x_2 > 0$, show $$\frac{\tan x_1}{x_1} > \frac{\tan x_2}{x_2}$$
We are given a triangle with the following property: one of its angles is quadrisected (divided into four equal angles) by the height, the angle bisector, and the median from that vertex. This property uniquely determines the triangle (up to scaling). Find the measure of the quadrisected angle.
Let $P$ be a point inside parallelogram $ABCD$. If $\angle{PAB}=\angle{PCB}$, show $\angle{PBA} = \angle{PDA}$.
Two sides of a triangle are 4 and 9; the median drawn to the third side has length 6. Find the length of the third side.
A right triangle has legs $a$ and $b$ and hypotenuse $c$. Two segments from the right angle to the hypotenuse are drawn,
dividing it into three equal parts of length $x=\frac{c}{3}$. If the segments have length $p$ and $q$, prove that $p^2 +q^2 =5x^2$.