In a right-angled triangle, the product of the other two sides (the two legs,…

In a right triangle with legs a, b and hypotenuse c, the Pythagorean identity a2 + b2 = c2 always holds. For any two positive reals, (a − b)2 ≥ 0, i.e. a2 + b2 ≥ 2ab, with equality exactly when a = b.

1. Let the two legs be a and b, and the hypotenuse be c. The given condition is ab = c2/2, i.e. 2ab = c2.

2. By the Pythagorean theorem, a2 + b2 = c2. Substituting c2 = 2ab gives a2 + b2 = 2ab.

3. Rearranging: a2 − 2ab + b2 = 0, i.e. (a − b)2 = 0.

4. This forces a = b, so the two legs are equal — the triangle is an isosceles right triangle.

5. In an isosceles right triangle the two acute angles are equal and sum to 90°, so each acute angle equals 45°.

Cross-check: with a = b = 1, c = √2, the product ab = 1 and c2/2 = 2/2 = 1 — the condition holds exactly, and tan(angle) = b/a = 1 gives angle = 45°, confirming the result.