Unit 4 test: Geometry

In triangle $ABC$, point $D$ is on $BC$ with $BD : DC = 5 : 2$, and point $E$ is on $AD$ with $AE : ED = 3 : 4$. Line $BE$ meets $AC$ at $F$. Then $\dfrac{AF}{FC} = \dfrac{m}{n}$ in lowest terms. Find $m + n$.

In triangle $ABC$ with $AC = 18$, point $E$ is on $AB$ with $AE : EB = 1 : 3$, and point $D$ is on $BC$ with $BD : DC = 2 : 1$. Line $ED$ meets line $AC$ at $F$. Find $CF$.

From a point $P$ outside circle $\omega$ with center $O$, a tangent touches $\omega$ at $T$ with $PT = 12$. A line through $P$ meets $\omega$ at $A$ and $B$ with $PA = 8$ and $PB > PA$, and this line is at distance $5$ from $O$. Find $OP^2$.

Circles with radii $15$ and $13$ have centers $14$ units apart and meet at $P$ and $Q$. A common external tangent touches the circles at $S$ and $T$, and line $PQ$ meets segment $ST$ at $M$. Find $MP \cdot MQ$.

Quadrilateral $ABCD$ is inscribed in a circle, with $AB = 2$, $BC = 5$, $CD = 10$ and $DA = 11$. Find $AC^2 + BD^2$.

In acute triangle $ABC$, $BC = 26$. The feet of the altitudes from $B$ and $C$ are $E$ and $F$, and $EF = 10$. The circumradius of triangle $ABC$ is $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

Triangle $ABC$ has $AB = 8$, $AC = 15$ and $\angle A = 60^\circ$. Its inradius is $r$, where $r^2 = \dfrac{m}{n}$ in lowest terms. Find $m + n$.

Triangle $ABC$ has $AB = 12$, $AC = 18$ and $BC = 20$. The bisector of $\angle A$ meets $BC$ at $D$. Find $AD^2$.

A pyramid has a square base, and all eight of its edges have length $10$. Its volume can be written as $\dfrac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime and $n$ is squarefree. Find $m + n + p$.

A right circular cone has base radius $5$ and height $12$. Sphere $S_1$ is inscribed in the cone (tangent to the base and the lateral surface). A smaller sphere $S_2$ is tangent to the lateral surface and externally tangent to $S_1$, and lies between $S_1$ and the apex. The radius of $S_2$ is $\dfrac{m}{n}$ in lowest terms. Find $m + n$.

An equilateral triangle has vertices $A = 1 + 2i$ and $B = 7 + 4i$ in the complex plane, and its third vertex $C$ has the larger imaginary part of the two possibilities. Then $|C|^2 = a + b\sqrt{c}$, where $a$, $b$, $c$ are positive integers and $c$ is squarefree. Find $a + b + c$.

A regular $12$-gon is inscribed in a circle of radius $2$. Find the sum of the squares of the lengths of all $66$ segments joining pairs of its vertices.

Circles with radii $3$ and $12$ are externally tangent. Their common external tangent lines meet at $P$. Find the length of a tangent segment from $P$ to the larger circle.

Circles $\omega_1, \omega_2, \omega_3, \dots$ lie inside a $60^\circ$ angle, each tangent to both sides and to the next circle, getting smaller toward the vertex. Circle $\omega_1$ has radius $12$. The total area of all the circles is $k\pi$. Find $k$.

Triangle $ABC$ has $AB = 13$, $BC = 14$ and $CA = 15$. The bisector of $\angle A$ meets $BC$ at $D$. The circle with diameter $AD$ meets $AB$ again at $X$ and $AC$ again at $Y$. Then $XY = \dfrac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime and $n$ is squarefree. Find $m + n + p$.