Polya's Triangle Space

In his book Mathematical Discovery George Polya discusses the region in the left hand panel bounded by the lines x≤y, y≤1 and x+y>1. Clearly, placing the large GOLD point at 1,1 forms an equilateral triangle. Suitably chosen points will determine isosceles triangles and right triangles. Can any shape triangle be formed by placing the large GOLD point somewhere in the region or on its boundary?