Simson line

In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. Simsonline.png

Proof

In the shown diagram, we draw additional lines $AP$ and $BP$. Then, we have cyclic quadrilaterals $ACBP$, $PC_1A_1B$, and $PB_1AC_1$. (more will be added)

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