What is the proof that equiangular triangle is also called equilateral triangle?
Label the triangle ABC.
Draw the bisector of angle A to meet BC at D.
Then in triangles ABD and ACD,
angle ABD = angle ACD (equiangular triangle)
angle BAD = angle CAD (AD is angle bisector)
so angle ADB = angle ACD (third angle of triangles).
Also AD is common.
So, by ASA, triangle ABD is congruent to triangle ACD
and therefore AB = AC.
By drawing the bisector of angle B, it can be shown that AB =
BC.
Therefore, AB = BC = AC ie the triangle is equilateral.