Indija TS

Question

In an acute triangle ABC, AD is the median, BE is the internal angle bisector and CF is the altitude, with D, E, F respectively on sides BC, CA, AB. If DEF is equilateral, prove that ABC is also equilateral. FIGURE_0

Accepted Answer

Observe that in the right triangle BFC, BD = DC = a/2. Hence DF = a/2 and this gives DE = EF = a/2. Now in triangle BEC, D is the midpoint of BC and DE = DB = DC. Hence BEC = 90^. Since BE bisects ABC, we conclude that BA = BC and CE = EA. In the right triangle CFA, E is the midpoint of AC. Hence EC = EA = EF = a/2. We conclude that CA = a = CB. We obtain AB = BC = CA. Hence ABC is equilateral.