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Right Triangle Equal Angles

Problem

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.


Solution

Consider the following diagram.


As CD is perpendicular to AB, triangle ACD is a right angle triangle. In which case, angle DAC = 90 minus 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.
(see http://www.qbyte.org/puzzles/puzzle13.html#p123)

Problem ID: 248 (27 Nov 2005)     Difficulty: 2 Star

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