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

Given: AD, BE and CF, the altitudes of triangle ABC are equal.

To do: Prove triangle ABC is  an equilateral triangle.

Solution:

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 ∠B=∠C.

This implies that

AC = AB   (sides opposite to equal angles).

Similarly,

AD=BE⇒∠B=∠A

⇒AC=BC

Together, we get AB=BC=AC

So, △ABC is equilateral.

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Updated on: 10-Oct-2022

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