Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that CBD = 50°. Point E is on side AB such that BCE = 60°. Find the measure of CED.

Solution

Copyright © 1996-2008 Alexander Bogomolny

This is a trigonometric solution (Solution 2 from [Knop]).

Denote the unknown angle CED as x. Then CDE = 160° - x.

Apply the Law of Sines in ΔCDE:

And in ΔBCE,

For, sin 2α = 2sin α · cos α.

Since ΔBCD is isosceles, CD = BC, which allows us to write a trigonometric equation:

This we are going to solve presently.

Using sin(180° - α) = sin(α),

Using the addition formula for sine and the one for cosine,

which simplifies to

or,

From which x = 30°.

Reference

1. C. Knop, Nine Solutions to One Problem, Kvant, 1993, no 6.

Copyright © 1996-2008 Alexander Bogomolny