If all three altitudes of a triangle are equal, prove that the triangle is equilateral.

Dear student

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

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