Outer Napoleon Triangle
See complete Problem 246 at:
gogeometry.com/problem/p246_napoleon_theorem_external_equilateral_triangle.htm
Level: High School, SAT Prep, College geometry
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Friday, February 6, 2009
Problem 246. Napoleon's Theorem I. External Equilateral triangles
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Create a equilateral triangle AB1A2 ( point A2 is inside triangle ABC)
ReplyDelete1. Consider 2 Triangles AA2C1 and ACB. AC1/AB=AA2/AC=1/SQRT(3) and angle A2AC1=angle CAB. these 2 triangles are similar.
2. From (1) we have C1A2=BC/SQRT(3)= A1C
3. 2 triangles B1A2C1 and B1CA1 are congruent ( B1A2=B1C, C1A2=A1C and angle B1A2C1=angle B1CA1)
4. So B1C1=B1A1 . Similarly with the same way we get A1B1=A1C1 and triangle A1B1C1 is equilateral.