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.^[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 $ABC$ .

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

[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.

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