The 80-80-20 Triangle Problem, Solution #7

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

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Copyright © 1996-2018 Alexander Bogomolny

This is Solution 6 from [Knop] and is due to Maria Gelband, a high school student at the time.

Let M be on the bisector of angle ∠BCD with the property that ∠ABM = 30°.

Since ΔBCD is isosceles (BC = CD), CM is the perpendicular bisector of BD so that, BM = DM. From the choice of M, ∠DBM = 60° implying that ΔBDM is equilateral.

Since BE is the bisector of ∠DBM it is also the perpendicular bisector of DM.

In ΔCDM, E is the intersection of the perpendicular bisector of DM and the bisector of the angle at C. Thus E belongs to the circumcenter of ΔCDM. In the circumcircle, two inscribed angles, CED and CMD are equal, but the latter is equal half of ∠BMD making ∠CED = 30°.

Reference

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

The 80-80-20 Triangle Problem

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