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 3 of 5: Derive the angle and side-ratio criterion
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cos⁡A=AB2+AC24AB⋅AC=14(ABAC+ACAB)\cos A=\frac{AB^2+AC^2}{4AB\cdot AC}=\frac14\left(\frac{AB}{AC}+\frac{AC}{AB}\right)
Detailed analysis

Apply the cosine rule in triangles ABDABD and ACDACD and use AD2=3CD2AD^2=3CD^2. The resulting equation is cos⁡A=(AB2+AC2)/(4AB AC)\cos A=(AB^2+AC^2)/(4AB\,AC), or the displayed symmetric expression in the ratio AB/ACAB/AC.