Editing Lemoine point

{{Short description|Intersection of the three symmedian lines of a triangle}}
{{distinguish|Lemoine Point Conservation Area}}
[[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]].

[[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>

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.
The [[Gergonne point]] of a triangle is the same as the symmedian point of the triangle's [[contact triangle]].<ref>{{citation
 | last1 = Beban-Brkić | first1 = J.
 | last2 = Volenec | first2 = V.
 | last3 = Kolar-Begović | first3 = Z.
 | last4 = Kolar-Šuper | first4 = R.
 | journal = Rad Hrvatske Akademije Znanosti i Umjetnosti
 | mr = 3100227
 | pages = 95–106
 | title = On Gergonne point of the triangle in isotropic plane
 | volume = 17
 | year = 2013}}.</ref>

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}}

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>

For the extension to an irregular tetrahedron see [[symmedian]].

== Notes ==
{{Notelist}}

== References ==
{{Reflist}}

== External links ==
* {{mathworld|id=SymmedianPoint|title=Symmedian Point}}


[[Category:Triangle centers]]