Symmedians

Triangle with side bisectors (black), angle bisectors (dashed) and symmetrians (red). L = symmetry point

Just as a side bisector of a triangle is also called the median, the mirror image of a side bisector at the corresponding angle bisector (i.e. at the angle bisector that starts from the same corner as the side bisector) is called a symmetrian . The term is an abbreviation for "symmetrical median", comes from the Greek and means "reflection on the center line".

The three symmedians of a triangle intersect at a point, the so-called Lemoine point ( Lemoine point ), which is also called Grebe point or symmetrian point . This can be proven with the help of Ceva's theorem.

The intersection of the symmedians is a non-canonical distinguished point of the triangle .

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 ).