Equilateral Triangle ?

Geometry Level 3

It's easy to see that equilateral triangle $ABC$ has the following property:

There is a point $P$ inside triangle $ABC$ such that $\angle{ABP}=\angle{BCP}=\angle{CAP}=30^\circ.$

If a given triangle satisfies this property, is it necessarily equilateral?

No Yes

2 solutions

Chung-En Hung
Aug 19, 2019

Central angle:60°
Inscribed angle:30°
Central angle=Inscribed angle*2

Yes I actually got a level 3 question correct wooo!!

Deep state of denial Burgert - 1 year, 9 months ago

Bruh it has a 50 percent chance. Not a great feat...

Krishna Karthik - 1 year, 2 months ago

At the time it had less

Deep state of denial Burgert - 1 year, 1 month ago

Very interesting solution with circles and the knowledge of inscribed angles. Also, I like how you dropped perpendiculars to see that $\alpha=60^\circ$ , $\zeta=60^\circ$ , and $\varepsilon=60^\circ$ .

hi bye - 1 year, 9 months ago

Chan Tin Ping
Aug 19, 2019

Actually I have post this question before. The explanation are under this question.

https://brilliant.org/problems/equilateral-triangle-6/?ref_id=1419727

Equilateral Triangle ?

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