Pedal 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. The line through these points is the Simson line of P, named for Robert Simson. The concept was first published, however, by William Wallace in 1797.

The converse is also true; if the three closest points to P on three lines are collinear, and no two of the lines are parallel, then P lies on the circumcircle of the triangle formed by the three lines. Or in other words, the Simson line of a triangle ABC and a point P is just the pedal triangle of ABC and P that has degenerated into a straight line and this condition constrains the locus of P to trace the circumcircle of triangle ABC.

The method of proof is to show that ${\displaystyle \angle NMP+\angle PML=180^{\circ }}$. ${\displaystyle PCAB}$ is a cyclic quadrilateral, so ${\displaystyle \angle PBN+\angle ACP=\angle PBA+\angle ACP=180^{\circ }}$. ${\displaystyle PMNB}$ is a cyclic quadrilateral (Thales' theorem), so ${\displaystyle \angle PBN+\angle NMP=180^{\circ }}$. Hence ${\displaystyle \angle NMP=\angle ACP}$. Now ${\displaystyle PLCM}$ is cyclic, so ${\displaystyle \angle PML=\angle PCL=180^{\circ }-\angle ACP}$. Therefore ${\displaystyle \angle NMP+\angle PML=\angle ACP+(180^{\circ }-\angle ACP)=180^{\circ }}$.

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