# Show that a triangle is equilateral

A circle crosses the sides of a triangle, dividing each of them into three equal parts. Prove that the triangle is equilateral.

I think that the best way is to show that $$\angle BAC = \angle ABC$$, and then $$\angle ABC = \angle ACB$$. Something like that and I would be very grateful if you could help me because I don't see how this can be done.

By the power of a point theorem, $$AD\cdot AE=AK\cdot AF$$
$$2 AD^2 = 2 AK^2$$
That means that $$AC = 3 AD = 3 AK = AB$$
Analogically $$BA=BC$$
• I think it is much simpler to write the second equation as $$\frac29 AB^2=\frac29AC^2.$$ – user May 18 '19 at 9:51