Question

AB, BE & CF are the altitudes of triangle ABC & equal to each other.
Prove that: $\Delta ABC$ is an equilateral triangle.

Answer 1

$ABC$ is a triangle with $AD,BE$ and $CF$ being altitudes.

Given that $AD=BE=CF$

Now,

area of $△ABC=\frac{1}{2}BC\times AD=\frac{1}{2}AC\times BE=\frac{1}{2}AB\times CF$

$\Rightarrow \frac{1}{2}BC\times AD=\frac{1}{2}AC\times BE=\frac{1}{2}AB\times CF\because AD=BE=CF\Rightarrow \frac{1}{2}BC=\frac{1}{2}AC=\frac{1}{2}AB\Rightarrow BC=AC=AB$

Therefore we have all three sides equal .

$\therefore △ABC$ is an equilateral triangle.