The bisector of the right angle of a right triangle is easily handled with linear equations. Orient the triangle on the usual (x,y) coordinate system as shown below.

|\ | \ | \ a | \ | / \ | / \ |/_____ \ C b

Then the bisector of the right angle at C has equation y=x, whereas the hypotenuse has equation x/b + y/a =1. These intersect at x=y=1/(1/a + 1/b)=a*b/(a+b). The length of the bisector is therefore [a*b/(a+b)]*sqrt(2).