Answer to Question #230 in Geometry for Todd

Question

The radius of the inscribed circle of the triangle equals 1, heights are integers. Prove that the triangle is equilateral.

Answer 1

In any triangle height is bigger than the diameter of inscribed circle. Thus heights of given triangle are integers bigger than 2, that's why all of them are not less than 3. Let S be the square of the triangle, a – its biggest side, h – corresponding height. Suppose that the triangle is not equatorial. Then its perimeter P is less than 3a. Thus 3a > P = Pr = 2S = ha, i.e. h < 3 – a contradiction.