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. 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.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:Proposition 4The method of proof is 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 (Thales' theorem), so ∠ P B N + ∠ N M P = 180 ∘ {displaystyle angle PBN+angle NMP=180^{circ }}  . Hence ∠ N M P = ∠ A C P {displaystyle angle NMP=angle ACP}  . Now P L C M {displaystyle PLCM}   is cyclic, so ∠ P M L = ∠ P C L = 180 ∘ − ∠ A C P {displaystyle angle PML=angle PCL=180^{circ }-angle ACP}  . Therefore ∠ N M P + ∠ P M L = ∠ A C P + ( 180 ∘ − ∠ A C P ) = 180 ∘ {displaystyle angle NMP+angle PML=angle ACP+(180^{circ }-angle ACP)=180^{circ }}  .