Consider a right triangle with an inscribed circle. Let x and y be the lengths of the two line segments formed on the hypotenuse by the point of tangency with the circle. What interesting fact can you prove about x*y?

The area A of the right triangle with inradius r may be expressed in two different ways:

A = inradius x perimeter / 2 =  r * [(x + r) + (y + r) + (x + y)] / 2  = r * (x + y + r), and

A = base x height / 2 = (x + r) * (y + r) / 2 =  r * (x + y + r) / 2 + x * y / 2 = A / 2 + x * y / 2.

Hence x * y = A.

Good problem, owl.  You are a wise one.  This is not a well-known area formula for a right triangle!

Edited to remove <br> format commands that were not needed here.

Edited on February 28, 2005, 7:01 pm

Posted by Richard on 2005-02-28 18:49:33