Simson line

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The Simson line LN (red) of the triangle ABC with respect to point P on the circumcircle Pedal Line.svg
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]

Contents

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.

Equation

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 [4] :Proposition 4

where an overbar indicates complex conjugation.

Properties

Simson lines (in red) are tangents to the Steiner deltoid (in blue). Simson-deltoid-anim.gif
Simson lines (in red) are tangents to the Steiner deltoid (in blue).

Proof of existence

The method of proof is to show that . is a cyclic quadrilateral, so . is a cyclic quadrilateral (Thales' theorem), so . Hence . Now is cyclic, so . Therefore .

Alternate proof

green line is the Simpson's line, the blue ones are the perpendiculars dropped. Simpson line.png
green line is the Simpson's line, the blue ones are the perpendiculars dropped.

Whatever the point Z in the adjacent figure is, a + c is 90. Also, whatever the point Z is, c and b are going to be equal. Therefore, we have the following:

a + c = 90

∴ a + b = 90 …(c and b are equal) (1)

Now, consider the measure of angle: a + 90 + b.

If we show that this angle is 180, then the Simpson’s theorem is proved.

From (1) we have, a + 90 + b = 180

Q.E.D.

Generalizations

Generalization 1

The projections of Ap, Bp, Cp onto BC, CA, AB are three collinear points A generalization of the Simson line.svg
The projections of Ap, Bp, Cp onto BC, CA, AB are three collinear points
  • Let ABC be a triangle, let a line ℓ go through circumcenter O, and let a point P lie on the circumcircle. Let AP, BP, CP meet ℓ at Ap, Bp, Cp respectively. Let A0, B0, C0 be the projections of Ap, Bp, Cp onto BC, CA, AB, respectively. Then A0, B0, C0 are collinear. Moreover, the new line passes through the midpoint of PH, where H is the orthocenter of ΔABC. If ℓ passes through P, the line coincides with the Simson line. [8] [9] [10]
A projective version of a Simson line A propjective Simson line.svg
A projective version of a Simson line

Generalization 2

  • Let the vertices of the triangle ABC lie on the conic Γ, and let Q, P be two points in the plane. Let PA, PB, PC intersect the conic at A1, B1, C1 respectively. QA1 intersects BC at A2, QB1 intersects AC at B2, and QC1 intersects AB at C2. Then the four points A2, B2, C2, and P are collinear if only if Q lies on the conic Γ. [11]

Generalization 3

See also

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References

  1. H.S.M. Coxeter and S.L. Greitzer, Geometry revisited, Math. Assoc. America, 1967: p.41.
  2. "Gibson History 7 - Robert Simson". 2008-01-30.
  3. http://www-groups.dcs.st-and.ac.uk/history/Biographies/Wallace.html
  4. Todor Zaharinov, "The Simson triangle and its properties", Forum Geometricorum 17 (2017), 373--381. http://forumgeom.fau.edu/FG2017volume17/FG201736.pdf
  5. Daniela Ferrarello, Maria Flavia Mammana, and Mario Pennisi, "Pedal Polygons", Forum Geometricorum 13 (2013) 153–164: Theorem 4.
  6. Olga Radko and Emmanuel Tsukerman, "The Perpendicular Bisector Construction, the Isoptic point, and the Simson Line of a Quadrilateral", Forum Geometricorum 12 (2012).
  7. Tsukerman, Emmanuel (2013). "On Polygons Admitting a Simson Line as Discrete Analogs of Parabolas" (PDF). Forum Geometricorum. 13: 197–208.
  8. "A Generalization of Simson Line". Cut-the-knot. April 2015.
  9. Nguyen Van Linh (2016), "Another synthetic proof of Dao's generalization of the Simson line theorem" (PDF), Forum Geometricorum, 16: 57–61
  10. Nguyen Le Phuoc and Nguyen Chuong Chi (2016). 100.24 A synthetic proof of Dao's generalisation of the Simson line theorem. The Mathematical Gazette, 100, pp 341-345. doi:10.1017/mag.2016.77. The Mathematical Gazette
  11. Smith, Geoff (2015), "99.20 A projective Simson line", The Mathematical Gazette, 99 (545): 339–341, doi:10.1017/mag.2015.47