Area in Equilateral Triangle

Point P lies inside an equilateral triangle ABC. Points D, E, F are the feet of the perpendiculars of point P onto the sides BC, CA, AB, respectively.
Prove that the sum of the colored areas is equal to the sum of the non-colored areas.

(Credits to Waldemar Pompe for introducing me to this problem)

(This has a simple, elegant solution)


Might not be your elegant solution but it’s the best I could come up with that doesn’t use algebra haha


Is this true for any regular polygon?

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This is unreal! @AwesomeLife_Math - this looks correct to me but you are the owner of this question so can control the :white_check_mark: button.

@mon1 - can you clarify for me how we know the first 3 triangles created are equilateral?

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Because the sides of the new smaller triangles are parallel to the big equilateral one so the size of the angles will be preserved. Ie angles of smaller triangles are still 60 so they are equilateral.

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Ah of course! Thanks :grin:

This was the solution I was looking for!
I’m not so sure about the “true for all regular polygons” thing.
It certainly is true for a square but things get a little complicated from a pentagon.