1. A triangle $A B C$ is given, right-angled at $B$. Prove that this triangle is isosceles if and only if the altitude $B H$ relative to the hypotenuse is congruent to half the hypotenuse.
1. A triangle $A B C$ is given, right-angled at $B$. Prove that this triangle is isosceles if and only if the altitude $B H$ relative to the hypotenuse is congruent to half the hypotenuse.