The 80-80-20 Triangle Problem, A Derivative, Solution #2

ABC is an isosceles triangle with vertex angle ∠BAC = 20° and AB = AC. Point E is on AB such that AE = BC. Find the measure of ∠AEC.

Solution

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Copyright © 1996-2018 Alexander Bogomolny

The solution is by Ian McGee of the University of Waterloo [Honsberger]. It was also found independently by Aditi Khetwal, a student of John Rhodes at the Woodhouse College, London, UK.

Construct an isosceles triangle AEQ with ∠AQE = 20°. Since. AE = BC, the latter is equal to ΔABC. In particular, AQ = AC. Also,

∠CAQ = ∠EAQ - ∠EAC = 80° - 20° = 60°.

Which makes ΔACQ equilateral. In particular, CQ = EQ = AQ.

In ΔCQE, EQ = CQ and ∠CQE = 60° - 20° = 40°. Thus, ∠CEQ = (180° - 40°) / 2 = 70° so that ∠AEC = 80° + 70° = 150°.

Reference

  1. R. Honsberger, Mathematical Chestnuts from Around the World, MAA, 2001, Ch 2.

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Copyright © 1996-2018 Alexander Bogomolny

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