Symmedian

In geometry, 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.

The three symmedians meet at a triangle center called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry".[1]

Isogonality

Many times in geometry, if we take three special lines through the vertices of a triangle, or cevians, then their reflections about the corresponding angle bisectors, 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.

  • In the diagram, the medians (in black) intersect at the centroid G.
  • Because the symmedians (in red) are isogonal to the medians, the symmedians also intersect at a single point, L.

This point is called the triangle's symmedian point, or alternatively the Lemoine point or Grebe point.

The dotted lines are the angle bisectors; the symmedians and medians are symmetric about the angle bisectors (hence the name "symmedian.")

Tetrahedra

The concept of a symmedian point extends to (irregular) tetrahedra. Given a tetrahedron ABCD two planes P and Q through AB are isogonal conjugates if they form equal angles with the planes ABC and ABD. Let M be the midpoint of the side CD. The plane containing the side AB that is isogonal to the plane ABM is called a symmedian plane of the tetrahedron. The symmedian planes can be shown to intersect at a point, the symmedian point. This is also the point that minimizes the squared distance from the faces of the tetrahedron.[2]

References

  1. Honsberger, Ross (1995), "Chapter 7: The Symmedian Point", Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Washington, D.C.: Mathematical Association of America.
  2. Sadek, Jawad; Bani-Yaghoub, Majid; Rhee, Noah (2016), "Isogonal Conjugates in a Tetrahedron" (PDF), Forum Geometricorum, 16: 43–50.
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