Concept

Lemoine point

Summary
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. 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(6). For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein. The symmedian point of a triangle with side lengths a, b and c has homogeneous trilinear coordinates [a : b : c]. 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 pr
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