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Tangent Circle in a Parallelogram

The tangent circle, perpendicular feet, and cyclic configurations used in the solution to Crux Mathematicorum Problem 5088.

Methods SVG construction

Parallelogram ABCD with BC equal to BD and M the midpoint of CD. E, F, and G are the perpendicular feet from M to the lines BD, AD, and BC. The circle centered at M passes through E, F, G, and K; AK is tangent at K, and N is the intersection of EG and BM.
The circle centered at M and the auxiliary points used in the proof.Select the diagram to enlarge it.

This independently prepared figure records only the configuration used in the accompanying solution. It does not reproduce the original problem statement.