Geometry Problem 1609: Radius of a Semicircle Tangent to Two Sides of a Triangle

Problem Statement

Let \( ABC \) be a triangle with sides of lengths \( a \), \( b \), and \( c \). A semicircle is constructed with its diameter on side \( AC \) and tangent to the other two sides, \( AB \) and \( BC \).

Prove that the radius \( r \) of the semicircle is given by

\( r = \dfrac{2\sqrt{s(s - a)(s - b)(s - c)}}{a + b} \)

where \( s = \dfrac{a + b + c}{2} \) is the semiperimeter of the triangle.