M CUBE: Math-e-Matics by Maheshwari INMO

Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.

INMO 2011 Solution 1: Solution 2: Here is a pure geometric solution. Consider the triangle BDF, CED and AFE with BD, CE and AF as bases. The sides DF, ED and FE make equal angles θ with the bases of respective triangles. If B ≥ C ≥ A, then it is easy to see that … Continue reading Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.