The length of the median of a right-angled triangle to the hypotenuse is 5 / √π

The length of the median of a right-angled triangle to the hypotenuse is 5 / √π, the perimeter of the triangle is 24 / √π, find the area of a circle inscribed in the triangle.

Given:
BM = 5 / √π
P (ABC) = 24 / √π

Find:
S (circle) -?

1) AC = 2 * BM = 2 * (5 / √π) = (10 / √π);
2) S (ABC) = p (ABC) * r = AB * BC / 2;
r = (AB * BC) / (2 * p (ABC)) = (AB * BC) / (2 * P (ABC) / 2) = AB * BC / P (ABC);
4) AC^2 = AB^2 + BC^2 = (10 / √π) 2;
AB + BC = P (ABC) – AC = (24 / √π) – (10 / √π) = (14 / √π);
(AB + BC) ^2 = AB^2 + BC^2 + 2 * AB * BC = AC2 + 2 * AB * BC;
(14 / √π) 2 = (10 / √π) 2 + 2 * AB * BC;
(196 / π) = (100 / π) + 2 * AB * BC;
2 * AB * BC = (96 / π);
AB * BC = (48 / π);
5) r = AB * BC / P (ABC) = (48 / π) / (24 / √π) = (2√π / π);
6) S (circle) = π * r2 = π * (2√π / π) 2 = π * 4π / π2 = 4.

Answer: The area of the circle is 4.