Lemoine point
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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.
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">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>Template:Citation.</ref>
The symmedian point of a triangle with side lengths Template:Mvar, Template:Mvar and Template:Mvar has homogeneous trilinear coordinates Template:Math.<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 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 Gergonne point of a triangle is the same as the symmedian point of the triangle's contact triangle.<ref>Template:Citation.</ref>
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.Template:Cn
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">Template:Citation.</ref>
For the extension to an irregular tetrahedron see symmedian.