A square is inscribed in a right-angled triangle, one of the corners of which coincides with the right angle

A square is inscribed in a right-angled triangle, one of the corners of which coincides with the right angle of the triangle. The sum of the lengths of the legs of the triangle is a, and their product is b. Find the length of the side of the square.

Triangles ABC and CEK are rectangular, in which the angle C is common, then triangles ABC and СEK are similar in acute angle.

Then AB / EK = AC / СK.

AB * СK = EK * AC.

СK = AC – AK = AC – EK

Then AB * (AC – EK) = EK * AC.

AB * AC – AB * EK = EK * AC.

AB * AC = AB * EK + EK * AC.

AB * AC = EK * (AB + AC).

EK = AB * BC / (AB + AC).

By condition, AB * BC = b, (AB + BC) = a, then EK = b / a.

Answer: The side of the square is b / a.