Iterations on Euler Lines
What is this about?
A Mathematical Droodle

13 July 2015, Created with GeoGebra

Let the Euler line of $\Delta ABC$ meet line $AB$ in $M$ and line $AC$ in $N.$ Then the Euler line of $\Delta AMN$ is parallel to $BC.$

13 July 2015, Created with GeoGebra

This is a particular case of a more general problem.

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Iterations on Euler LinesWhat is this about? A Mathematical Droodle