Right Triangle Equal Angles
In the triangle ABC, C is a right angle, CD is perpendicular to AB, and C" is the midpoint of AB and C is joined to C".
Prove that α = γ.
Consider the following diagram.
As CD is perpendicular to AB, triangle ACD is a right angle triangle. In which case, angle DAC = 90 α.
We are given that angle ACD is a right angle, so in triangle ABC we can see that angle ABC = α.
As any triangle in a semi-circle is a right angle, C"A = C"B = C"C (radii) and we deduce that triangle BC"C is isosceles. Therefore angle C"BC = angle C"CB; that is, α = γ. QED
Prove the converse: if α = γ then angle ACB must be a right angle.