Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Find the answer

Let $A B C$ be a triangle and $D, E$ , and $F$ be the midpoints of sides $B C, C A$ , and $A B$ respectively. What is the maximum number of circles which pass through at least 3 of these 6 points?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

All $\binom{6}{3}=20$ triples of points can produce distinct circles aside from the case where the three points are collinear $(B D C, C E A, A F B)$ .

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