Parallelogram with Side Lines through Fixed Points
What Might This Be About?
If the four side lines of a parallelogram pass through four collinear fixed points, the diagonals, too, pass through fixed points on the same line:
Denote the point involved as below:
By Thales' theorem,
which means that $M$ divides $EG$ in the ratio $HG:FE.$ Since points $E,F,F,H$ are fixed so is that ratio and, therefore, point $M.$ Point $N$ is fixed for the same reason.
In Exercices de Géométrie (Jacques Gabay, 1991, #1124, Theorem 322) F. G.-M. credits E. Catalan with this statement.
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