261. Given the three sides of a right triangle, determine the radius of the circle inscribed in this triangle.
261. Let the sides of triangle $ABC$ be given (Fig. 60): the hypotenuse $AC=b$; the legs $AB=c$ and $BC=a$. Clearly, if we inscribe a circle of radius $r$ in the triangle, we will have:

$$
AB = AE + r \quad BC = CF + r
$$

or

$$
c + a = AE + CF + 2r
$$

But

$$
AE = AD \text{ and } CF = CD \text{. }
$$

Adding gives:

$$
AE + CF = AD + DC = AC = b
$$

Therefore:

from which

$$
c + a = b + 2r
$$

$$
r = \frac{a + c - b}{2}
$$