Simson line

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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 1799, and is sometimes called the Wallace line. 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. Placing the triangle in the complex plane, let the triangle ABC with unit circumcircle have vertices whose locations have complex coordinates a, b, c, and let P with complex coordinates p be a point on the circumcircle. The Simson line is the set of points z satisfying where an overbar indicates complex conjugation. It suffices to show that ∠ N M P + ∠ P M L = 180 ∘ {\displaystyle \angle NMP+\angle PML=180^{\circ }} . P C A B {\displaystyle PCAB} is a cyclic quadrilateral, so ∠ P B A + ∠ A C P = ∠ P B N + ∠ A C P = 180 ∘ {\displaystyle \angle PBA+\angle ACP=\angle PBN+\angle ACP=180^{\circ }} . P M N B {\displaystyle PMNB} is a cyclic quadrilateral (since ∠ P M B = ∠ P N B = 90 ∘ {\displaystyle \angle PMB=\angle PNB=90^{\circ }} ), so ∠ P B N + ∠ N M P = 180 ∘ {\displaystyle \angle PBN+\angle NMP=180^{\circ }} .