Equilateral triangles (marked); M, N, P are midpoints of AD, CF, EB. Prove MNP is...

[Taric]

The red triangle, the green one and the blue one are equilaterals.
M, N and P are the midpoints of AD, CF and EB.
Prove that MNP is equilateral.

 

[Deleted member 4993]

The red triangle, the green one and the blue one are equilaterals.
M, N and P are the midpoints of AD, CF and EB.
Prove that MNP is equilateral.

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[Taric]

I've found that AOE and BOF are similar triangles. Hence AE = BF and BEFA is an isoscele trapezoid.

 

[Taric]

Trapezoid

AOE and BOF are similar triangles. Hence ABEF is an isosceles trapezoid. And that's all I could find

 

[Taric]

What else can you say about these 3 triangles?

There is nothing relevant to say about them

 

[Taric]

They are also congruent.

Yes, they are

 

[yma16]

proof

The red triangle, the green one and the blue one are equilaterals.
M, N and P are the midpoints of AD, CF and EB.
Prove that MNP is equilateral.

 

[Taric]

Very impressive!

View attachment 9651View attachment 9652

Thanks a lot for this solution.
Meantime, I've found another one, using complex numbers (I mean affixes)