Lemoine point: Difference between revisions
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[[File:Lemoine punkt.svg|thumb|upright=1.25|A triangle with [[Median (geometry)|medians]] (black), [[angle bisectors]] (dotted) and [[symmedian]]s (red). The symmedians intersect in the symmedian point L, the angle bisectors in the [[incenter]] I and the medians in the [[centroid]] G.]] | [[File:Lemoine punkt.svg|thumb|upright=1.25|A triangle with [[Median (geometry)|medians]] (black), [[angle bisectors]] (dotted) and [[symmedian]]s (red). The symmedians intersect in the symmedian point L, the angle bisectors in the [[incenter]] I and the medians in the [[centroid]] G.]] | ||
In [[geometry]], the '''Lemoine point''', '''Grebe point''' or '''symmedian point''' is the intersection of the three [[symmedian]]s ([[Median (geometry)|medians]] reflected at the associated [[angle bisectors]]) of a triangle. In other words, it is the [[isogonal conjugate]] of the [[centroid]]. | In [[geometry]], the '''Lemoine point''', '''Grebe point''' or '''symmedian point''' is the intersection of the three [[symmedian]]s ([[Median (geometry)|medians]] reflected at the associated [[angle bisectors]]) of a triangle. In other words, it is the [[isogonal conjugate]] of the [[centroid]] of a triangle. | ||
[[Ross Honsberger]] called its existence "one of the crown jewels of modern geometry".<ref name="h"/> | [[Ross Honsberger]] called its existence "one of the crown jewels of modern geometry".<ref name="h"/> | ||
In the [[Encyclopedia of Triangle Centers]] the symmedian point appears as the sixth point, X(6).<ref name="etc">[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers], accessed 2014-11-06.</ref> For a non-equilateral triangle, it lies in the open [[orthocentroidal disk]] punctured at its own center, and could be any point therein.<ref>{{citation|first1=Christopher J.|last1=Bradley|first2=Geoff C.|last2=Smith|title=The locations of triangle centers|journal=Forum Geometricorum|volume=6|year=2006|pages=57–70|url=http://forumgeom.fau.edu/FG2006volume6/FG200607index.html|access-date=2016-10-18|archive-date=2016-03-04|archive-url=https://web.archive.org/web/20160304000426/http://forumgeom.fau.edu/FG2006volume6/FG200607index.html|url-status=dead}}.</ref> | In the [[Encyclopedia of Triangle Centers]] the symmedian point appears as the sixth point, X(6).<ref name="etc">[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers], accessed 2014-11-06.</ref> For a non-equilateral triangle, it lies in the open [[orthocentroidal disk]] punctured at its own center, and could be any point therein.<ref>{{citation|first1=Christopher J.|last1=Bradley|first2=Geoff C.|last2=Smith|title=The locations of triangle centers|journal=Forum Geometricorum|volume=6|year=2006|pages=57–70|url=http://forumgeom.fau.edu/FG2006volume6/FG200607index.html|access-date=2016-10-18|archive-date=2016-03-04|archive-url=https://web.archive.org/web/20160304000426/http://forumgeom.fau.edu/FG2006volume6/FG200607index.html|url-status=dead}}.</ref> | ||
== Construction == | |||
The symmedian point of a triangle with side lengths {{mvar|a}}, {{mvar|b}} and {{mvar|c}} has homogeneous [[trilinear coordinates]] {{math|[''a'' : ''b'' : ''c'']}}.<ref name="etc"/> | The symmedian point of a triangle with side lengths {{mvar|a}}, {{mvar|b}} and {{mvar|c}} has homogeneous [[trilinear coordinates]] {{math|[''a'' : ''b'' : ''c'']}}.<ref name="etc"/> | ||
An algebraic way to find the symmedian point is to express the triangle by three linear equations in two unknowns given by the [[hesse normal form]]s of the corresponding lines. The solution of this [[overdetermined system]] found by the [[least squares method]] gives the coordinates of the point. It also solves the optimization problem to find the point with a minimal sum of squared distances from the sides. | An algebraic way to find the symmedian point is to express the triangle by three linear equations in two unknowns given by the [[hesse normal form]]s of the corresponding lines. The solution of this [[overdetermined system]] found by the [[least squares method]] gives the coordinates of the point. It also solves the optimization problem to find the point with a minimal sum of squared distances from the sides. | ||
The [[Gergonne point]] of a triangle is the same as the symmedian point of the | |||
The symmedian point of a triangle {{mvar|ABC}} can be constructed in the following way: let the [[Tangent|tangent lines]] of the circumcircle of {{mvar|ABC}} through {{mvar|B}} and {{mvar|C}} meet at {{mvar|A'}}, and analogously define {{mvar|B'}} and {{mvar|C'}}; then {{mvar|A'B'C'}} is the [[tangential triangle]] of {{mvar|ABC}}, and the lines {{mvar|AA'}}, {{mvar|BB'}} and {{mvar|CC'}} intersect at the symmedian point of {{mvar|ABC}}.{{efn|If ABC is a right triangle with right angle at A, this statement needs to be modified by dropping the reference to AA' since the point A' does not exist.}} It can be shown that these three lines meet at a point using [[Brianchon's theorem]]. Line {{mvar|AA'}} is a symmedian, as can be seen by drawing the circle with center {{mvar|A'}} through {{mvar|B}} and {{mvar|C}}.{{cn|date=February 2016}} | |||
== Relation to other triangle centers == | |||
The [[Gergonne point]] of a triangle is the same as the symmedian point of the [[contact triangle]].<ref>{{citation | |||
| last1 = Beban-Brkić | first1 = J. | | last1 = Beban-Brkić | first1 = J. | ||
| last2 = Volenec | first2 = V. | | last2 = Volenec | first2 = V. | ||
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| volume = 17 | | volume = 17 | ||
| year = 2013}}.</ref> | | year = 2013}}.</ref> | ||
The [[mittenpunkt]] of a triangle is the same as the symmedian point of the [[excentral triangle]]. | |||
The | The centroid of the [[pedal triangle]] of the symmedian point is the symmedian point.<ref name="h"/> The centroid of the antipedal triangle of the symmedian point is the [[circumcenter]].<ref>{{Cite web |last=Humenberger |first=Hans |title=Finding Triangles With Given Circum-medial Triangle |url=https://homepage.univie.ac.at/hans.humenberger/Aufsaetze/Steiner-inellipse-gazette-online-appendix.pdf |url-status=live |access-date=27 June 2025 |website=Homepage Hans HUMENBERGER}}</ref> | ||
== Tetrahedra == | |||
For the extension to an irregular tetrahedron see [[symmedian]]. | For the extension to an irregular tetrahedron see [[symmedian]]. | ||
== History == | |||
The French mathematician [[Émile Lemoine]] proved the existence of the symmedian point in 1873, and [[Ernst Wilhelm Grebe]] published a paper on it in 1847. [[Simon Antoine Jean L'Huilier]] had also noted the point in 1809.<ref name="h">{{citation|first=Ross|last=Honsberger|authorlink=Ross Honsberger|contribution=Chapter 7: The Symmedian Point|title=Episodes in Nineteenth and Twentieth Century Euclidean Geometry|publisher=[[Mathematical Association of America]]|location=Washington, D.C.|year=1995}}.</ref> | |||
== Notes == | == Notes == | ||
Latest revision as of 10:01, 27 June 2025
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In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle. In other words, it is the isogonal conjugate of the centroid of a triangle.
Ross Honsberger called its existence "one of the crown jewels of modern geometry".[1]
In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X(6).[2] For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.[3]
Construction
The symmedian point of a triangle with side lengths Template:Mvar, Template:Mvar and Template:Mvar has homogeneous trilinear coordinates Template:Math.[2]
An algebraic way to find the symmedian point is to express the triangle by three linear equations in two unknowns given by the hesse normal forms of the corresponding lines. The solution of this overdetermined system found by the least squares method gives the coordinates of the point. It also solves the optimization problem to find the point with a minimal sum of squared distances from the sides.
The symmedian point of a triangle Template:Mvar can be constructed in the following way: let the tangent lines of the circumcircle of Template:Mvar through Template:Mvar and Template:Mvar meet at Template:Mvar, and analogously define Template:Mvar and Template:Mvar; then Template:Mvar is the tangential triangle of Template:Mvar, and the lines Template:Mvar, Template:Mvar and Template:Mvar intersect at the symmedian point of Template:Mvar.Template:Efn It can be shown that these three lines meet at a point using Brianchon's theorem. Line Template:Mvar is a symmedian, as can be seen by drawing the circle with center Template:Mvar through Template:Mvar and Template:Mvar.Script error: No such module "Unsubst".
Relation to other triangle centers
The Gergonne point of a triangle is the same as the symmedian point of the contact triangle.[4] The mittenpunkt of a triangle is the same as the symmedian point of the excentral triangle.
The centroid of the pedal triangle of the symmedian point is the symmedian point.[1] The centroid of the antipedal triangle of the symmedian point is the circumcenter.[5]
Tetrahedra
For the extension to an irregular tetrahedron see symmedian.
History
The French mathematician Émile Lemoine proved the existence of the symmedian point in 1873, and Ernst Wilhelm Grebe published a paper on it in 1847. Simon Antoine Jean L'Huilier had also noted the point in 1809.[1]
Notes
References
External links
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