Question

Question #48167

Answer 1

Let's take triangle ABC, where side a is opposite A, side b is opposite B, and side c is opposite C. Now assume this triangle has altitudes of d, e, and f, and these 3 altitudes are congruent.

Now construct equilateral triangle LMN with opposite sides l, m, and n, just like ABC. Construct altitude o that intersects side l and vertex L. Keep in mind that since all 3 sides are equal and the triangle is equiangular, if you rotate this triangle, it will map onto itself. This means that the altitude we drew will map onto the other 2 altitudes exactly, so they must all be congruent.

Since this is true, we can assume the converse, which means that triangle ABC must be equilateral.