1990 AIME Problem 14

The rectangle ABCD^{}_{} below has dimensions AB^{}_{} = 12 \sqrt{3} and BC^{}_{} = 13 \sqrt{3}. Diagonals \overline{AC} and \overline{BD} intersect at P^{}_{}. If triangle ABP^{}_{} is cut out and removed, edges \overline{AP} and \overline{BP} are joined, and the figure is then creased along segments \overline{CP} and \overline{DP}, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.