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Prove that the triangle is equilateral

Source: Mediterranean MO 2004

October 31, 2010
geometrycircumcircletrigonometrycomplex numbersgeometry proposed

Problem Statement

Let z1,z2,z3z_1, z_2, z_3 be pairwise distinct complex numbers satisfying ∣z1∣=∣z2∣=∣z3∣=1|z_1| = |z_2| = |z_3| = 1 and 12+∣z1+z2∣+12+∣z2+z3∣+12+∣z3+z1∣=1.\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1. If the points A(z1),B(z2),C(z3)A(z_1),B(z_2),C(z_3) are vertices of an acute-angled triangle, prove that this triangle is equilateral.
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