Question

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.)

Answer 1

Given: ∠A = ∠B = ∠C

To Prove: AB = BC = CA

Proof:

In ΔABC:
∠A = ∠C (Given)

AB = BC ... (1) (Sides opposite to equal angles are equal)

∠B = ∠A (Given)

BC = CA ... (2) (Sides opposite to equal angles are equal)

∠B = ∠C (Given)

AC = AB ... (3) (Sides opposite to equal angles are equal)

From (1), (2) and (3):

AB = BC = CA

Thus, the given triangle is equilateral.