Lemoine point

A triangle with medians (blue), angle bisectors (green) and symmedians (red). The symmedians intersect in the symmedian point L, the angle bisectors in the incenter I and the medians in the centroid G.

The symmedian point, Lemoine point or Grebe 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".[1]

In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X(6).[2] It lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.[3]

The symmedian point of a triangle with side lengths a, b and c has homogeneous trilinear coordinates [a : b : c].[2]

The Gergonne point of a triangle is the same as the symmedian point of the triangle's contact triangle.[4]

The symmedian point of a triangle ABC can be constructed in the following way: let the tangent lines of the circumcircle of ABC through B and C meet at A', and analogously define B' and C'; then A'B'C' is the tangential triangle of ABC, and the lines AA', BB' and CC' intersect at the symmedian point of ABC.[5] It can be shown that these three lines meet at a point using Brianchon's theorem. Line AA' is a symmedian, as can be seen by drawing the circle with center A' through B and C.

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.[1]

For the extension to an irregular tetrahedron see symmedian.

References

  1. 1 2 Honsberger, Ross (1995), "Chapter 7: The Symmedian Point", Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Washington, D.C.: Mathematical Association of America .
  2. 1 2 Encyclopedia of Triangle Centers, accessed 2014-11-06.
  3. ↑ Bradley, Christopher J.; Smith, Geoff C. (2006), "The locations of triangle centers", Forum Geometricorum, 6: 57–70 .
  4. ↑ Beban-Brkić, J.; Volenec, V.; Kolar-Begović, Z.; Kolar-Šuper, R. (2013), "On Gergonne point of the triangle in isotropic plane", Rad Hrvatske Akademije Znanosti i Umjetnosti, 17: 95–106, MR 3100227 .
  5. ↑ If ABC is a right triangle with right angle at A, this statement needs to be modified by dropping the reference to AA' since the point A' does not exist.
  • Weisstein, Eric W. "Symmedian Point". MathWorld.
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