Lemoine point

Lemoine point L as the intersection of the symmedians (red)

The Lemoine point of a triangle , also called Lemoine point , Grebe point or symmetry point , is an excellent point in the triangle . It is the intersection of the side bisectors mirrored at the bisector, the symmedians .

properties

• The Lemoine point is by definition isogonally conjugated to the centroid .
• If we denote the triangle with ABC and the Lemoine point with  L , then the distances of the point  L to the straight lines BC , CA and AB are proportional to the lengths of the sides BC , CA and AB of the triangle ABC .
• The Lemoine point is the solution of an occasionally important optimization problem : If we consider a point  P in the plane of the triangle ABC , then the sum of the squares of the distances from the point P to the sides BC , CA and AB is minimal if and only if P with the Lemoine point  L of triangle ABC coincides.
• This optimization problem is also dissolved or Lemoinepunkt found when the three sides of the triangle by three linear equations of the respective straight lines in Hesse normal form are expressed in two variables and a balancing solution of the overdetermined linear system of equations using the method of least squares is determined .
• The Lemoine point of the larger triangle, which is determined by the three circle centers, is the so-called center point of the triangle.

Coordinates

Lemoine point (Symmedian point, Grebe point, ) ${\displaystyle X_{6}}$
Trilinear coordinates	${\displaystyle a\,:\,b\,:\,c=\,\sin \alpha \,:\,\sin \beta \,:\,\sin \gamma }$
Barycentric coordinates	${\displaystyle a^{2}\,:\,b^{2}\,:\,c^{2}}$

history

The point is named in England and France after the French mathematician Émile Lemoine and in Germany also after the German mathematician Ernst Wilhelm Grebe , both of whom published about him. However, the point was known before their publications.

literature

• Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 213, 268, 271, 303 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ).

Web links

• Eric W. Weisstein : Symmedian Point . In: MathWorld (English).
• X6 The Symmedian Point. An extensive presentation with various further statements