Symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle of the triangle there in two equal parts). The three symmedians intersect in a single point, the triangle's symmedian point or Lemoine point or Grebe point, the latter names coming from Émile Lemoine, the French mathematician who proved its existence in 1873, and Ernst Wilhelm Grebe who published a paper on it 1847. Simon Antoine Jean L'Huilier had also noted the point in 1809.
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The symmedian point of a triangle with sides a, b and c has homogeneous trilinear coordinates [a : b : c].
The Gergonne point of a triangle is the same as the symmedian point of the triangle's contact triangle.
The symmedian point is the isogonal conjugate of the triangle's centroid.