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 4 of 5: Rule out angles greater than 60 degrees
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ABAC+ACAB≥2⇒cos⁡A≥12⇒A≤60∘\frac{AB}{AC}+\frac{AC}{AB}\ge2\Rightarrow\cos A\ge\frac12\Rightarrow A\le60^\circ
Detailed analysis

By AM–GM, the sum of a positive ratio and its reciprocal is at least 22. Thus the criterion requires cos⁡A≥1/2\cos A\ge1/2, so no triangle exists when A>60∘A>60^\circ.