The legs of a right triangle are equal to A and B. Find the bisector of a right angle.

1. Let the bisector of the right angle be equal to X. It divides the triangle into two triangles with sides. In one triangle, sides A, X and the angle between them are 45 degrees, and in another triangle, sides B, X and the angle between them are 45 degrees.
2. The area of ​​a right-angled triangle is equal to the sum of the areas of two triangles. We find the areas of two triangles through the sine of the angle:
S1 = (A * X * sin 45) / 2;
S2 = (B * X * sin 45) / 2;
3. The area of ​​a right-angled triangle is:
S = (A * B) / 2;
4. We get the following equation:
(A * X * sin 45) / 2 + (B * X * sin 45) / 2 = (A * B) / 2;
sin 45 = 1 / √ 2;
X = (A * B * √ 2) / (A + B);
5. Answer: (A * B * √ 2) / (A + B).