Isosceles Triangles Equal in Perimeter and Area

Beginning with the observation that to every non-equilateral, isosceles triangle, there is a unique, noncongruent isosceles triangle with the same perimeter and the same area, we determine all such pairs with integer sides and integer areas. We also determine those pairs in which the sides in each isosceles triangle are relatively prime. Finally, we give an infinite family of pairs of triangles equal in perimeter and in area, one isosceles and one non-isosceles.