1. Let P be the centroid of triangle ABC. If triangle ABP is an equilateral triangle with a side length of 2, then find the perimeter of triangle ABC.
2. In triangle PQR, the side lengths are PQ = 9, PR = 10, and QR = 17. Let X be the intersection of the angle bisector of angle P with side QR, and let Y be the foot of the perpendicular from X to side PR. Compute the length of side XY.
3. A circle is centered at O and has an area of 48pi. Let Q and R be points on the circle, and let P be the circumcenter of triangle QRO. If P is contained in triangle QRO, and triangle PQR is equilateral, then find the area of triangle PQR.
4. In triangle ABC, point P is on side BC such that PA = 13, PB = 14, PC = 4, and the circumcircles of triangles APB and APC have the same radius. Find the area of triangle ABC.