Question

In a right triangle ABC, $\angle B$ is a right angle and [BD] is an altitude drawn to the hypotenuse [AC]. Prove that |BD| is equal to the sum of the radii of the circles inscribed in the triangles ABC, ADB, and CDB.

Answer 1

We know that, in a right angled triangle, radius of incircle $r_{ABC}=\frac{AB+BC-AC}{2}$

Also, ABD is a right triangle with AB as hypotenuse. Its inradius is given by $r_{ABD}=\frac{BD+AD-AB}{2}$

Also, CDB is a right triangle with CB as hypotenuse. Its inradius is given by $r_{CDB}=\frac{CD+BD-CB}{2}$

$r_{ABC}+r_{ADB}+r_{CDB}=\frac{AB+BC-AC+BD+AD-AB+CD+BD-CB}{2}$

$=\frac{2BD+AD+CD-AC}{2}=BD$ since $AC=AD+DC$