Question

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

Answer 1

consider triangle ABE and triangle ACF

BE = CF (given)

angle A = angle A (common)

angle AEB = angle AFC = 90 degrees

therefore by AAS congruency triangle ABE is congruent to triangle ACF

AB = AC (c.p.c.t.)

similarly, triangle DAC is congruent to triangle CBE

AC = BC (c.p.c.t.)

similarly, triangle BCF is congruent to triangle ABD

AB = BC (c.p.c.t.)

now,

AB = BC

AC = BC

AB = AC

this implies that

AB = BC = AC

Hence, triangle ABC is an equilateral triangle.