Question

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

Prove it.

Answer 1

Given,

AD, BE and CF are the altitudes drawn on sides BC, CA and AB of $\Delta$ ABC such that AD = BE = CF

Area of $\Delta ABC=\frac{1}{2}xBC\times AD=\frac{1}{2}\times AB\times CF=\frac{1}{2}\times CA\times BE$

(Since, Area of $\Delta =\frac{1}{2}\times Base\times Correspondingaltitude$

∴ BC × AD = AB × CF = CA × BE

BC = AB = CA (Since, AD = BE = CF)

Hence, $\Delta$ ABC is an equilateral triangle.