Michael Kleber writes:

http://www.mathpuzzle.com/triangle.html <http://www.mathpuzzle.com/triangle.html> says "As it happens, a [7, 7, 13] triangle can be divided into two non-similar triangles with integer sides: [3, 5, 7] and [3, 7, 8]", but does not mention the relation between the two. Boggle.

It does, see http://www.mathpuzzle.com/ITKK.gif there. Leo