Quote:
Assume an equalatral triangle. (The test said I could)

I had hoped that this was the trick, but sadly P can be found for at least some isoceles triangles -- a 6/5/5 triangle has P (degenerately) in the centre of the 6 side (common perimeter 12), and a 10/13/13 triangle has P 5/9 of the way along the symmetry line (common perimeter 80/3).

The construction is fairly straightforward for the isoceles case; the hard bit of this question is either constructing it for scalene, or proving that impossible.

Peter