Question 12
In isosceles triangle $ABC,$ $A$ is located at the origin and $B$ is located at $(20, 0).$ Point $C$ is in the first quadrant with $AC = BC$ and $\angle BAC = 75^\circ.$ If $\triangle ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive $y$-axis, the area of the region common to the original triangle and the rotated triangle is in the form $p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s,$ where $p,$ $q,$ $r,$ $s$ are integers. Find $\frac{p - q + r - s}{2}.$