Problem 1

In triangle

ABC

, let

D,E,F

be the midpoints of

BC,CA,AB

, respectively, and let

G

be the centroid. For each value of

\angle BAC

, how many non-similar triangles are there for which

AEGF

is a cyclic quadrilateral?

Step 5 of 5: Count the non-similar triangles

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A=60^\circ: AB=AC;\qquad A<60^\circ: r+\frac1r=4\cos A\text{ has two reciprocal roots}

Detailed analysis

For $A=60^\circ$ , equality in AM–GM forces $AB=AC$ , giving one equilateral similarity class. For $A<60^\circ$ , put $r=AB/AC$ ; then $r+1/r=4\cos A>2$ has two positive reciprocal roots. Interchanging $B,C$ exchanges the roots and produces similar triangles, so there is exactly one non-similar triangle. Therefore the answer is $0$ for $A>60^\circ$ and $1$ for every $A\le60^\circ$ .