Question

If all the three altitudes of a triangle are equal, the triangle is equilateral. Prove it.

Answer 1

In right ΔBEC and ΔBFC,
BE = CF ........[Given]
BC = BC ........[Common]
∠BEC = ∠BFC ........[each = 90°]
∴ ΔBEC ≅ ∆CFB .........[RHS]
⇒ ∠B = ∠C
Similarly,
∠A = ∠B
Hence, ∠A = ∠B = ∠C
⇒ AB = BC = AC
Therefore, ABC is an equilateral triangle.