The legs of an isosceles right-angled triangle are 36 + 18√2. Find the radius of the circle inscribed in this triangle.

By the condition of the problem, in an isosceles right-angled triangle, legs a and b are 36 + 18√2.
Find the area S of this triangle:
S = a * b / 2 = (36 + 18√2) * (36 + 18√2) / 2 = 18 ^ 2 * (2 + √2) ^ 2/2 = 162 * (4 + 2√2 + 2) = 162 * (6 + 2√2) = 324 (3 + √2).
Using the Pythagorean theorem, we find the hypotenuse from a given triangle:
c = √ ((36 + 18√2) ^ 2 + (36 + 18√2) ^ 2) = √ (2 * (36 + 18√2) ^ 2) = √ (2 * (18 * (2 + √2) ^ 2)) = √ (2 * 18 ^ 2 * (6 + 2√2)) = √ (2 ^ 2 * 18 ^ 2 * (3 + √2)) = 2 * 18 * √ (3 + √2) = 36 * √ (3 + √2).
Using the formula S = r * (a + b + c) / 2, where r is the radius of the circle inscribed in the triangle, we find r:
r = S / ((a + b + c) / 2) = 2 * S / (a ​​+ b + c) = 2 * 324 * (3 + √2) / (36 + 18√2 + 36 + 18√ 2 + 36 * √ (3 + √2)) = 648 * (3 + √2) / (36 * (2 + √2 + √ (3 + √2))) = 18 * (3 + √2) / (2 + √2 + √ (3 + √2)).

Answer: the radius of a circle inscribed in a triangle is 18 * (3 + √2) / (2 + √2 + √ (3 + √2)).