The 80-80-20 Triangle
Several triangles due to their specific properties served as a source for research, wonderment, and a variety of problems. Some, like equilateral, right isosceles, golden, Egyptian, 30-60-90, are very well known. One triangle that deserves more recognition has probably entered mathematical folklore some 100 years ago. Tom Rike of Berkley Mathematical Circles mentions its appearance in the Mathematical Gazette, Volume 11 (1922), p. 173. (His online article also provides additional references.) This is the 80-80-20 (or sometimes 20-80-80) triangle, i.e., the isosceles triangle with the apex angle of 20° and the base angles of 80°.
The original problem gave rise to a few modifications; and each of them has been solved in many, many ways. I'll be adding solutions and perhaps problems related to the original one. Any assistance is welcomed.
This page is to serve as the portal for this undertaking.
Original problem
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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.
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Solutions
60-70 Variant
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Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that CBD = 60°. Point E is on side AB such that BCE = 70°. Find the measure of CED.
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Solutions
Short segments equality
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ABC is an isosceles triangle with vertex angle BAC = 20° and AB = AC. Draw BCD = 60°; D lying on AB. Draw an arc with B as center and radius equal to BC. Let this arc cut AC at point E and AB at the point F. Prove that CE = DF.
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Solutions
Long segments equality
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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.
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Solutions
Partition into adjacent isosceles triangles
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The 80-80-20 triangle can be partitioned into isosceles triangle with bases on the legs of the given triangle. For what other apex angles A can isosceles triangle ABC be tessellated with isosceles triangles in a similar manner?
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Solutions
Partition that starts with a base angle bisector
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The 80-80-20 triangle can be partitioned in a curious way: let BD, with D on AC, bisect angle B. Then AD = BC + BD.
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Solutions
40-50 Variant
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Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that CBD = 40°. Point E is on side AB such that BCE = 50°. Find the measure of CED.
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Solutions
30-40 Variant
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Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that CBD = 30°. Point E is on side AB such that BCE = 40°. Find the measure of CED.
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Solutions
20-30 Variant
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Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that CBD = 20°. Point E is on side AB such that BCE = 30°. Find the measure of CED.
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Solutions
10-20 Variant
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Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that CBD = 10°. Point E is on side AB such that BCE = 20°. Find the measure of CED.
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Solutions
References
- T. Rike, An Intriguing Geometry Problem, Berkeley Math Circle, May 5, 2002.
- H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, MAA, 1967.
- R. Honsberger, Four Minor Gems from Geometry, Mathematical Gems II, MAA, 1976.
- R. Honsberger, Three Solutions to a Variation on an Old Chestnut, Mathematical Chestnuts from Around the World, MAA, 2001.
- C. Knop, Nine Solutions to One Problem, Kvant, 1993, no 6.
- R. Leikin, Dividable Triangles — What Are They?, Mathematics Teacher, May 2001, pp. 392–398.
- V. V. Prasolov, Essays on Numbers and Figures, MAA, 2000.
Copyright © 1996-2010 Alexander Bogomolny
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