What is the side of the square?

A hard-level "Length" problem.If the legs are a and b, the inscribed square's side is always a×b÷(a+b). Here 10×15÷25 = 6.

Hints

Call the top-right corner of the square Q. Look carefully: Q sits exactly on the hypotenuse. That is the clue.
A small right triangle appears above the square. Let's focus on this blue triangle.
The blue triangle and the original big one share the right angle and the top angle. All angles match, so they are similar.
The original triangle is 10 cm tall and 15 cm wide: height : width = 10 : 15 = 2 : 3. Being similar, the blue one has the same ratio.
The blue triangle's height is 10 cm minus the square, i.e. (10 − ?), and its width is the square's side, ?.
Writing it as a ratio. Multiply the height side by 3 and the width side by 2.
Tidying up: 5 lots of ? make 30. Now just divide.

Answer 6cm