Lemoine point
In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians 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".
In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X. For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.
Construction
The symmedian point of a triangle with side lengths, and has homogeneous trilinear coordinates.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 can be constructed in the following way: let the tangent lines of the circumcircle of through and meet at, and analogously define and ; then is the tangential triangle of, and the lines, and intersect at the symmedian point of. It can be shown that these three lines meet at a point using Brianchon's theorem. Line is a symmedian, as can be seen by drawing the circle with center through and.
Relation to other triangle centers
The Gergonne point of a triangle is the same as the symmedian point of the contact triangle.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. The centroid of the antipedal triangle of the symmedian point is the circumcenter.