Symmedians are three particular geometrical

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 lines associated with every triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

. They are constructed by taking a median

Median (geometry)

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side...

 of the triangle (a line connecting a vertex

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

 with the midpoint

Midpoint

The midpoint is the middle point of a line segment. It is equidistant from both endpoints.-Formulas:...

 of the opposite side), and reflecting the line over the corresponding angle bisector

Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector In geometry, bisection is the division of something into two equal...

 (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 or Grebe point, the latter names coming from Émile Lemoine

Émile Lemoine

Émile Michel Hyacinthe Lemoine was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most notably, the École Polytechnique...

, the French mathematician who proved its existence in 1873, and Ernst Wilhelm Grebe who published a paper on it 1847. Simon Antoine Jean L'Huilier

Simon Antoine Jean L'Huilier

Simon Antoine Jean L'Huilier was a Swiss mathematician of French Hugenot descent...

 had also noted the point in 1809.

Particular points

The symmedian point of a triangle with sides a, b and c has homogeneous trilinear coordinates

Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous 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

Isogonal conjugate

In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C. These three reflected lines concur at the isogonal conjugate of P...

 of the triangle's centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

.

External links

• Symmedian and Antiparallel at cut-the-knot
  Cut-the-knot
  Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
  
  
• Symmedian and 2 Antiparallels at cut-the-knot
  Cut-the-knot
  Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
  
  
• Symmedian and the Tangents at cut-the-knot
  Cut-the-knot
  Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
  
  
• An interactive Java applet for the symmedian point
• Symmedian Point article at Wolfram MathWorld
• Isogons and Isogonic Symmetry