Two rays from the right-angled vertex of an isosceles right triangle split the hypotenuse into three sgments.

Prove that the three segments can be used to form a right triangle (the middle segment as the hypotenuse) if and only if the angle between the rays is 45°.

Let the triangle have a right angle at O and, wlog let the hypotenuse have length 2 units and mid-point M. Let A and B be at distances a and b from M, measured in opposite directions along the hypotenuse.

The three segment lengths can form the sides of a right angled triangle

            <=>     (a + b)² =  (1 - a)²  +  (1 - b)²

            <=>         2ab   =  2 - 2a - 2b

            <=>     (a + b)/(1 - ab)  =  1

            <=>     (tanAOM + tanBOM)/(1 - tanAOM * tanBOM) = 1

            <=>     tan AOB  =  1

            <=>     AOB = 45⁰

Edited on April 5, 2011, 7:20 pm

Posted by Harry on 2011-04-05 19:13:08