Question

if all three altitudes of a triangle are equal then prove that it 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.

Answer 2

Answer:

Given,

AD, BE and CF are altitudes on sides BC, CA and AB of Δ ABC respectively 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 proved

Hence, ΔABC is an equilateral triangle.

Hope it helps :D