If all three altitudes of a triangle are equal, prove that the triangle is equilateral.
Given, AD, BE and CF are the attitudes drawn on sides BC, CA and AB of Δ ABC such that AD = BE = CF
Area of Δ ABC = × BC × AD = × AB × CF = × CA × BE (Area of Δ = × Base × Correspondence attitude)
∴ BC × AD = AB × CF = CA × BE
⇒ BC = AB = CA ( AD = BE = CF)
Hence, ΔABC is an equilateral triangle.Regards