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 symmedian point. Ross Honsberger 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 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; the symmedians and medians are symmetric about the angle bisectors (hence the name "symmedian.")
Symmedian point
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.[citation needed]
History
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]
References
- ^ a b Honsberger, Ross (1995), "Chapter 7: The Symmedian Point", Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Washington, D.C.: Mathematical Association of America.
- ^ a b Encyclopedia of Triangle Centers, accessed 2014-11-06.
- ^ Bradley, Christopher J.; Smith, Geoff C. (2006), "The locations of triangle centers", Forum Geometricorum, 6: 57–70.
- ^ 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.
- ^ 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.