Practice (42)

3306

Let $CD$ be the altitude in right $\triangle{ABC}$ from the right angle $C$. If inradii of $\triangle{ABC}$, $\triangle{ACD}$, and $\triangle{BCD}$ be $r_1$, $r_2$, and $r_3$, respectively, show that $$r_1 + r_2 + r_3 = CD$$

3307

Three circles are tangent to each other and also a common line, as shown. Let the radii of circles $O_1$, $O_2$, and $O_3$ be $r_1$, $r_2$, and $r_3$, respectively. Show that $$\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_1}} +\frac{1}{\sqrt{r_2}}$$

3308

Given two segments $AB$ and $MN$, show that $$MN\perp AB \Leftrightarrow AM^2 - BM^2 = AN^2 - BN^2$$

3309

Let point $P$ inside an equilateral $\triangle{ABC}$ such that $AP=3$, $BP=4$, and $CP=5$. Find the side length of $\triangle{ABC}$.

3310

Let $M$ be a point inside $\triangle{ABC}$. Draw $MA'\perp BC$, $MB'\perp CA$, and $MC'\perp AB$ such that $BA'=BC'$ and $CA'=CB'$. Prove $AB'=AC'$.

3613

Find one root to $\sqrt{3}x^7 + x^4 + 2=0$.

3866

(Apollonius’ Theorem) Let $AD$ be one median of $\triangle{ABC}$ where point $D$ lies on side $BC$. Show that the following relation holds: $$AB^2 +AC^2 = 2\times(AD^2 +BD^2)$$

3907

Compute $\sin 15^\circ$ and $\cos 15^\circ$ using a geometry approach.

3934

In $\triangle{ABC}$, let $\angle{A}=120^\circ$. If $A'$, $B'$ and $C'$ are feet of the three interior angle bisectors as shown, prove $A'B'\perp A'C'$.

3936

As shown, both $ABCD$ and $OPRQ$ are squares. Additionally, $O$ is the center of $ABCD$, $OP=1$, $BP=\sqrt{2}$, and $CQ=\sqrt{5}$. Find the length of $DR$.

4104

In $\triangle ABC, AB = AC = 10$ and $BC = 12$. Point $D$ lies strictly between $A$ and $B$ on $\overline{AB}$ and point $E$ lies strictly between $A$ and $C$ on $\overline{AC}$ so that $AD = DE = EC$. Then $AD$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

4108

Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.

4119

In equiangular octagon $CAROLINE$, $CA = RO = LI = NE =$ $\sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, the total area of the six triangular regions. Then $K =$ $\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.

4129

The incircle $\omega$ of triangle $ABC$ is tangent to $\overline{BC}$ at $X$. Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$. Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$. Assume that $AP = 3$, $PB = 4$, $AC = 8$, and $AQ = \dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

4273

Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle{ABC}$ respectively. Show that $OH\parallel BC$ if and only if $\tan{B}\tan{C}=3$.

4743

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$ ?

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