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{{short description|Line constructed from a triangle}}
[[Image:Pedal Line.svg|right|thumb|250px|The Simson line ''LN'' (red) of the triangle {{mvar|ABC}} with respect to point {{mvar|P}} on the circumcircle]]

In [[geometry]], given a [[triangle]] {{mvar|ABC}} and a [[Point (geometry)|point]] {{mvar|P}} on its [[circumcircle]], the three closest points to {{mvar|P}} on lines {{mvar|AB}}, {{mvar|AC}}, and {{mvar|BC}} are [[collinear]].<ref>H.S.M. Coxeter and S.L. Greitzer, ''Geometry revisited'', Math. Assoc. America, 1967: p.41.</ref> The line through these points is the '''Simson line''' of {{mvar|P}}, named for [[Robert Simson]].<ref>{{cite web|url=http://www-groups.dcs.st-and.ac.uk/Extras/Gibson_history_7.html|title=Gibson History 7 - Robert Simson|date= 2008-01-30|work=[[MacTutor History of Mathematics archive]]}}</ref> The concept was first published, however, by [[William Wallace (mathematician)|William Wallace]] in 1799,<ref>{{cite web|url=http://www-groups.dcs.st-and.ac.uk/Biographies/Wallace.html|title=William Wallace|work=MacTutor History of Mathematics archive}}</ref> and is sometimes called the '''Wallace line'''.<ref>{{cite journal |last=Clawson |first=J. W. |title=A Theorem in the Geometry of the Triangle |journal=The American Mathematical Monthly |year=1919 |volume=26 |number=2 |pages=59–62 |jstor=2973140 }}</ref>

The [[Theorem#Converse|converse]] is also true; if the three closest points to {{mvar|P}} on three lines are collinear, and no two of the lines are parallel, then {{mvar|P}} lies on the circumcircle of the triangle formed by the three lines. Or in other words, the Simson line of a triangle {{mvar|ABC}} and a point {{mvar|P}} is just the [[pedal triangle]] of {{mvar|ABC}} and {{mvar|P}} that has degenerated into a straight line and this condition constrains the [[Locus (mathematics)|locus]] of {{mvar|P}} to trace the circumcircle of triangle {{mvar|ABC}}.

==Equation==

Placing the triangle in the complex plane, let the triangle {{mvar|ABC}} with unit [[circumcircle]] have vertices whose locations have complex coordinates {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, and let P with complex coordinates {{mvar|p}} be a point on the circumcircle. The Simson line is the set of points {{mvar|z}} satisfying<ref name=TZ>Todor Zaharinov, "The Simson triangle and its properties", ''Forum Geometricorum'' 17 (2017), 373--381. http://forumgeom.fau.edu/FG2017volume17/FG201736.pdf</ref>{{rp|Proposition 4}}

:<math>2abc\bar{z} -2pz+p^2+(a+b+c)p -(bc+ca+ab)-\frac{abc}{p} =0,</math>

where an overbar indicates [[complex conjugation]].

==Properties==

[[Image:Simson-deltoid-anim.gif|right|thumb|250px|Simson lines (in red) are tangents to the Steiner [[Deltoid curve|deltoid]] (in blue).]]

*The Simson line of a vertex of the triangle is the [[altitude (geometry)|altitude]] of the triangle dropped from that vertex, and the Simson line of the point [[diametrically opposite]] to the vertex is the side of the triangle opposite to that vertex.
*If {{mvar|P}} and {{mvar|Q}} are points on the circumcircle, then the angle between the Simson lines of {{mvar|P}} and {{mvar|Q}} is half the angle of the arc {{mvar|PQ}}. In particular, if the points are diametrically opposite, their Simson lines are perpendicular and in this case the intersection of the lines lies on the [[nine-point circle]].
*Letting {{mvar|H}} denote the [[orthocenter]] of the triangle {{mvar|ABC}}, the Simson line of {{mvar|P}} [[Bisection|bisects]] the segment {{mvar|PH}} in a point that lies on the nine-point circle.
*Given two triangles with the same circumcircle, the angle between the Simson lines of a point {{mvar|P}} on the circumcircle for both triangles does not depend of {{mvar|P}}.
*The set of all Simson lines, when drawn, form an [[Envelope (mathematics)|envelope]] in the shape of a deltoid known as the [[Steiner deltoid]] of the reference triangle.
*The construction of the Simson line that coincides with a side of the reference triangle (see first property above) yields a nontrivial point on this side line. This point is the reflection of the foot of the altitude (dropped onto the side line) about the midpoint of the side line being constructed. Furthermore, this point is a tangent point between the side of the reference triangle and its Steiner deltoid.
* A [[quadrilateral]] that is not a [[parallelogram]] has one and only one pedal point, called the Simson point, with respect to which the feet on the quadrilateral are collinear.<ref>[http://forumgeom.fau.edu/FG2013volume13/FG201316.pdf Daniela Ferrarello, Maria Flavia Mammana, and Mario Pennisi, "Pedal Polygons", ''Forum Geometricorum'' 13 (2013) 153–164: Theorem 4.]</ref> The Simson point of a [[trapezoid]] is the point of intersection of the two nonparallel sides.<ref>Olga Radko and Emmanuel Tsukerman, "The Perpendicular Bisector Construction, the Isoptic point, and the Simson Line of a Quadrilateral", ''Forum Geometricorum'' 12 (2012). [http://forumgeom.fau.edu/FG2012volume12/FG201214.pdf]</ref>{{rp|p. 186}}
* No [[convex polygon]] with at least 5 sides has a Simson line.<ref>{{cite journal | last1 = Tsukerman | first1 = Emmanuel | year = 2013 | title = On Polygons Admitting a Simson Line as Discrete Analogs of Parabolas | url = http://forumgeom.fau.edu/FG2013volume13/FG201321.pdf | journal = Forum Geometricorum | volume = 13 | pages = 197–208 }}</ref>

==Proof of existence==
It suffices to show that <math>\angle NMP + \angle PML = 180^\circ</math>.

<math>PCAB</math> is a cyclic quadrilateral, so <math>\angle PBA + \angle ACP = \angle PBN + \angle ACP = 180^\circ</math>. <math>PMNB</math> is a cyclic quadrilateral (since <math>\angle PMB=\angle PNB = 90^\circ</math>), so <math>\angle PBN + \angle NMP = 180^\circ</math>. Hence <math>\angle NMP = \angle ACP</math>. Now <math>PLCM</math> is cyclic, so <math>\angle PML = \angle PCL = 180^\circ - \angle ACP</math>.

Therefore <math>\angle NMP + \angle PML = \angle ACP + (180^\circ - \angle ACP) = 180^\circ</math>.

==Generalizations==

===Generalization 1===
[[File:A generalization of the Simson line.svg|thumb|250px|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 ''A<sub>p</sub>, B<sub>p</sub>, C<sub>p</sub>'' respectively. Let ''A''<sub>0</sub>, ''B''<sub>0</sub>, ''C''<sub>0</sub> be the projections of ''A<sub>p</sub>, B<sub>p</sub>, C<sub>p</sub>'' onto ''BC, CA, AB'', respectively. Then ''A''<sub>0</sub>, ''B''<sub>0</sub>, ''C''<sub>0</sub> are collinear. Moreover, the new line passes through the midpoint of ''PH'', where ''H'' is the orthocenter of Δ''ABC''. If ℓ passes through ''P'', the line coincides with the Simson line.<ref>{{cite web|url=http://www.cut-the-knot.org/m/Geometry/GeneralizationSimson.shtml|title=A Generalization of Simson Line|publisher=Cut-the-knot|date = April 2015}}</ref><ref>{{citation|url=https://forumgeom.fau.edu/FG2016volume16/FG201608.pdf|archive-url=https://web.archive.org/web/20231023035726/https://forumgeom.fau.edu/FG2016volume16/FG201608.pdf|author= Nguyen Van Linh|title= Another synthetic proof of Dao's generalization of the Simson line theorem|journal= Forum Geometricorum|volume= 16|year=2016|pages= 57–61|archive-date=2023-10-23}}</ref><ref name=NguyenLePhuocandNguyenChuongChi>[http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=10362951&fileId=S0025557216000772 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]]</ref>

[[File:A propjective Simson line.svg|thumb|right|250px|A projective version of a Simson line]]

===Generalization 2===

* Let the vertices of the triangle ''ABC'' lie on the [[conic section|conic]] Γ, and let ''Q, P'' be two points in the plane. Let ''PA, PB, PC'' intersect the conic at ''A''<sub>1</sub>, ''B''<sub>1</sub>, ''C''<sub>1</sub> respectively. ''QA''<sub>1</sub> intersects ''BC'' at ''A''<sub>2</sub>, ''QB''<sub>1</sub> intersects ''AC'' at ''B''<sub>2</sub>, and ''QC''<sub>1</sub> intersects ''AB'' at ''C''<sub>2</sub>. Then the four points ''A''<sub>2</sub>, ''B''<sub>2</sub>, ''C''<sub>2</sub>, and ''P'' are collinear if only if ''Q'' lies on the conic Γ.<ref>{{citation|url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9834854&fulltextType=XX&fileId=S0025557215020549|first= Geoff|last= Smith|year=2015|title= 99.20 A projective Simson line|journal= The Mathematical Gazette|volume= 99|issue=545|pages=339–341|doi= 10.1017/mag.2015.47|s2cid= 124965348|url-access= subscription}}</ref>

===Generalization 3===
* R. F. Cyster generalized the theorem to [[cyclic quadrilateral]]s in [https://www.jstor.org/stable/3606490 The Simson Lines of a Cyclic Quadrilateral]

==See also==
*[[Longuerre's theorem]]
*[[Pedal triangle]]
*[[Robert Simson]]

==References==
<references />

==External links==
{{Commons category|Simson line}}
* [http://www.cut-the-knot.org/Curriculum/Geometry/Simpson.shtml Simson Line] at [[cut-the-knot]].org
* F. M. Jackson and {{mathworld | urlname = SimsonLine | title = Simson Line}}
* [http://dynamicmathematicslearning.com/miquel.html A generalization of Neuberg's theorem and the Simson-Wallace line] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], an interactive dynamic geometry sketch.
* [https://www.geogebra.org/m/y9q44rty Simson line] at geogebra.org (interactive illustration)
* [https://users.sch.gr/kodulis/geo/neo.php?r=../quiz/g/j/depo/th283.js&t=y&l=0 Simson line], [https://users.sch.gr/kodulis/geo/neo.php?r=../quiz/g/j/depo/th303.js&t=y&l=0 Simson lines of antidiametric poles - Euler circle], [https://users.sch.gr/kodulis/geo/neo.php?r=../quiz/g/j/depo/th305.js&t=y&l=0 Simson line - Euler circle], [https://users.sch.gr/kodulis/geo/neo.php?r=../quiz/g/j/depo/th304.js&t=y&l=0 Simson line paralleles] at [https://users.sch.gr/kodulis/geo/?l=0 Interactive Geometry]

[[Category:Straight lines defined for a triangle]]