The legs of a right triangle are 7 and 24 find the distance between the centers of the inscribed and circumscribed circles.

A = 7, b = 24.
c = √ (a ^ 2 + b ^ 2) = √ (7 ^ 2 + 24 ^ 2) = 25.
The radius of the circumscribed circle is half the hypotenuse. R = c / 2 = 12.
Inscribed circle radius: r = S / p.
S = ab / 2 = 724/2 = 84.
p = (a + b + c) / 2 = (7 + 24 + 25) / 2 = 28.
r = 84/28 = 3.
Euler’s formula: d ^ 2 = R ^ 2-2Rr, where d is the distance between the centers of the inscribed and circumscribed circles.
d ^ 2 = 12 ^ 2 – 2 12 3 = 72.
d = 6√2