Among right-angled triangles of equal height, which one has the smallest hypotenuse? (By height, we naturally mean the height drawn from the right-angle vertex to the hypotenuse.)
We know that in a right-angled triangle, the height is the geometric mean of the two segments of the hypotenuse.

$$
m=\sqrt{c_{1} c_{2}}
$$

According to the arithmetic and geometric mean inequality, the hypotenuse $c=c_{1}+c_{2} \geq 2 \sqrt{c_{1} c_{2}}=2 m$.

$c$ will be minimal when $c=2 m$, and this happens if and only if $c_{1}=c_{2}$, meaning the right-angled triangle is isosceles. Durst Endre

Solved by: Dancs I., Gerencsér Ottilia, Kántor S., Kovács L., Reichlin V., Tar D., Villányi O., Zatykó L., Zobor E. Using differential calculus: Gerencsér Ottilia.

Complicatedly: Rédly E.