MathLabs

Problem 1

In triangle ABCABC, let D,E,FD,E,F be the midpoints of BC,CA,ABBC,CA,AB, respectively, and let GG be the centroid. For each value of ∠BAC\angle BAC, how many non-similar triangles are there for which AEGFAEGF is a cyclic quadrilateral?
Step 2 of 5: Translate cyclicity into a median relation
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AEGF cyclic⟺AI⋅IG=FI⋅IE⟺AD2=3CD2AEGF\text{ cyclic}\Longleftrightarrow AI\cdot IG=FI\cdot IE\Longleftrightarrow AD^2=3CD^2
Detailed analysis

The intersecting-chords criterion at II says AEGFAEGF is cyclic exactly when AI⋅IG=FI⋅IEAI\cdot IG=FI\cdot IE. Substituting the ratios above gives the equivalent relation AD2=3CD2AD^2=3CD^2 (the form used in the official solution).