Question

In given figure the altitudes AD, BE and CF of triangle ABC are equal. Prove that ABC is an equilateral triangle.

Answer 1

In right-angle triangles BCE and CBF, we have,

BC = BC (common hypotenuse);

BE = CF (given).

Hence BCF and CBF are congruent, by RHS theorem. Comparing the triangles, we get $\angle B=\angle C$.

This implies that

AC = AB (sides opposite to equal angles).

Similarly,

$AD=BE\Rightarrow \angle B=\angle A$

$\Rightarrow AC=BC$

Together, we get $AB=BC=AC$or $△ABC$ is equilateral. $\left[henceproved\right]$