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 there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector.
Many times in geometry, if we take three special lines through the vertices of a triangle, or cevians, then their reflections about the angle bisector, called isogonal lines, will also have interesting properties. For instance, if three cevians of a triangle intersect at a point P, then their isogonal lines also intersect at a point, called the isogonal conjugate of P.
The symmedians illustrate this fact nicely.
- In the diagram, the medians (in blue) intersect at the centroid G.
- Because the symmedians (in red) are isogonal to the medians, the symmedians also intersect at a single point, K.
This point is called the triangle's symmedian point, or alternatively the Lemoine point or Grebe point.
The green lines are the angle bisectors; note how the symmedians and medians are symmetric about the angles bisectors (hence the name "symmedian.")
The French mathematician Émile Lemoine proved the existence of the symmedian point in 1873, and Ernst Wilhelm Grebe published a paper on it 1847. Simon Antoine Jean L'Huilier had also noted the point in 1809.
In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X_6.
The symmedian point of a triangle with sides a, b and c has homogeneous trilinear coordinates [a : b : c].
- Ross Honsberger, "The Symmedian Point," Chapter 7 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry, The Mathematical Association of America, Washington, D.C., 1995.