Chapter 3.P, Problem 16P

(a) Let ABC be a triangle with right angle A and hypotenuse $a=\left|BC\right|$.(see the figure.)

If the inscribed circle touches the hypotenuse at $D$, show that

$\left|CD\right|=\frac{1}{2}\left(\left|BC\right|+\left|AC\right|-\left|AB\right|\right)$

(b) If $\theta =\frac{1}{2}\angle C$ express the radius $r$ of the inscribed circle in terms of $a$ and $\theta$.

(c) If $a$ is fixed and $\theta$ varies, find the maximum value of $r$.