1.
In triangle $ABC$, $AB = AC,$ and angle $A$ is equal to $36^\circ.$ Point $D$ is on $\overline{AC}$ so that $\overline{BD}$ bisects $\angle ABC.$
(a) Prove that $BC = BD = AD$.
(b) Let $x = BC$ and let $y = CD$. Using similar triangles $BCD$ and $ABC$, write an equation involving $x$ and $y$.
(c) Let $r = \frac{y}{x}$. Write the equation from Part (b) in terms of $r,$ and find $r.$
(d) Find $\cos 36^\circ$ and $\cos 72^\circ$ using Parts a-c. (Do not use your calculator!)
2.
Let $a,$ $b,$ and $c$ be nonnegative real numbers, and let $A=(0,0),$ $B=(a,b),$ and $C=(c,0)$ be points in the coordinate plane, such that $AB=AC.$ Let $D$ be the midpoint of $\overline{BC},$ let $E$ be the foot of the altitude from $D$ to $\overline{AC},$ and let $F$ be the midpoint of $\overline{DE}.$
(a) Express the coordinates of points $E$ and $F$ in terms of $a,$ $b,$ and $c.$
(b) Show that line segments $\overline{AF}$ and $\overline{BE}$ are perpendicular.
3.
In right triangle $ABC,$ $AC = BC$ and $\angle C = 90^\circ.$ Let $P$ and $Q$ be points on hypotenuse $\overline{AB},$ as shown below, such that $\angle PCQ = 45^\circ.$ Show that
\[AP^2 + BQ^2 = PQ^2.\]
