Question

The altitudes of triangle ABC are equal . prove that triangle ABC is an equilateral triangle

Answer 1

Given,
 AD, BE and CF are the altitudes drawn on sides BC, CA and AB of Δ ABC such that AD = BE = CF
Area of Δ ABC = 1/2 x BC × AD = 1/2 × AB × CF = 1/2 × CA × BE
(Since, Area of Δ = 1/2 × Base × Correspondence altitude)
∴ BC × AD = AB × CF = CA × BE
BC = AB = CA (Since, AD = BE = CF)
Hence, ΔABC is an equilateral triangle.
Hence proved.