Structure of phosphorus pentachloride
The short answer is stated on the Wikipedia page for trigonal bipyramidal:


  … there is no geometrical arrangement with five terminal atoms in equivalent positions.


This website from the School of Mathematics and Statistics at UNSW explains why distributing points on a sphere is tough and that perfect distribution is possible only in certain cases $(n=1,2,4,6,8,12,\&amp;\ 20)$. These cases have been known for a long time — they correspond to the platonic solids (plus a point and a line segment). Even trigonal planar does not represent an even distribution on the sphere.

Trigonal bipyamidal then just happens to be the minimum energy conformation. The repulsive interactions between the chlorine atoms cannot be optimized but they can be minimized.