Suppose ABC is an equiangular triangle. Prove that it is equilateral.(You have seen earlier that an equilateral triangle is equiangular. Thus for triangles equiangularity is equivalent to equilaterality.)

Δ ABC is an equiangular triangle

Let AD be the perpendicular from A on BC

In Δ ABD and Δ ACD

∠ABD = ∠ACD (ΔABC is equiangular)

∠ADB = ∠ADC (AD is a perpendicular)

AD = AD (Common side)

So Δ ABD and Δ ACD are congruent to each other by A.A.S. axiom of congruency

AB = AC (Corresponding Parts of Congruent Triangles)

BD = DC(Corresponding Parts of Congruent Triangles)

Since the triangle is equiangular and AB = AC , so

AB = AC = BC

Hence the triangle is equilateral