Simson line

The Simson line LN (red) of the triangle ABC with respect to point P on the circumcircle

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.[1] The line through these points is the Simson line of P, named for Robert Simson.[2] The concept was first published, however, by William Wallace in 1799,[3] and is sometimes called the Wallace line.[4]

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.

  1. ^ H.S.M. Coxeter and S.L. Greitzer, Geometry revisited, Math. Assoc. America, 1967: p.41.
  2. ^ "Gibson History 7 - Robert Simson". MacTutor History of Mathematics archive. 2008-01-30.
  3. ^ "William Wallace". MacTutor History of Mathematics archive.
  4. ^ Clawson, J. W. (1919). "A Theorem in the Geometry of the Triangle". The American Mathematical Monthly. 26 (2): 59–62. JSTOR 2973140.