Thai Mathematical Olympiad

Question

Let A, B, C be three distinct points on a unit circle. Let G and H be the centroid and the orthocenter of the triangle ABC, respectively. Let F be the midpoint of the segment GH. Evaluate |AF|^2 + |BF|^2 + |CF|^2.

Accepted Answer

Define a coordinate system with the origin at the center of the circle. We can see that H = A + B + C and G = 13(A + B + C). Thus, F = G + H2 = 23(A + B + C). We now have aligned |AF|^2 + |BF|^2 + |CF|^2 &= (A - F) (A - F) + (B - F) (B - F) + (C - F) (C - F) &= |A|^2 + |B|^2 + |C|^2 - 2(A + B + C) F + 3F F &= |A|^2 + |B|^2 + |C|^2 - (2(A + B + C) - 3F) F &= |A|^2 + |B|^2 + |C|^2 = 3. aligned