Symmedian

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, the latter name coming from Émile Lemoine, the French mathematician who proved its existence.

Particular points

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.

References

* 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.

External links

* [http://www.cut-the-knot.org/Curriculum/Geometry/SymAntiparallel.shtml Symmedian and Antiparallel] at cut-the-knot
* [http://www.cut-the-knot.org/Curriculum/Geometry/Sym2Antiparallel.shtml Symmedian and 2 Antiparallels] at cut-the-knot
* [http://www.cut-the-knot.org/Curriculum/Geometry/Symmedian.shtml Symmedian and the Tangents] at cut-the-knot
* [http://www.uff.br/trianglecenters/X0006.html An interactive Java applet for the symmedian point]