Question

AD and BE and CF are altitudes of ∆ABC .If AD =,BE= CF prove ABC is equilateral triangle

Answer 1

consider triangle ABE and triangle ACF

BE = CF (given)

angle A = angle A (common)

angle AEB = angle AFC = 90degrees

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

therefore the triangle is equilateral