Problem 1 of Asian Pacific Mathematical Olympiad 1992

A triangle with sides a, b, and c is given. Denote by s the semiperimeter, that is s = (a+b+c)/2. Construct a triangle with sides s-a, s-b, and s-c. This process is repeated until a triangle can no longer be constructed with the side lengths given. For which original triangles can this process be repeated indefinitely?

Solution by Vo Duc Dien:

For the process to be repeated indefinitely the new triangle must be similar to the existing triangle; i.e., its angles must equal those of the original triangle. And one of those new triangles to satisfy that condition have its vertices at the midpoints of the sides of the original, and their sides equaled half the corresponding sides of the original triange.

We have: (a + b + c)/2 - a = b/2

           (a + b + c)/2 - b = c/2

           (a + b + c)/2 - c = a/2

Solve those three equations with three unknowns we have a = b = c. So the original triangle is equilateral.