geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians ( medians reflected at the associated angle bisectors) of a triangle. In other words, it is the

isogonal conjugate __NOTOC__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...

of the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

. Ross Honsberger called its existence "one of the crown jewels of modern geometry". In the

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. This resource is hosted at the University of Evansville The University of Evansville (UE) is a priv ...

the symmedian point appears as the sixth point, X(6).Encyclopedia of Triangle Centers
accessed 2014-11-06. For a non-equilateral triangle, it lies in the open

orthocentroidal disk In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter. This diameter also contains the triangle's nine-point center and is a subset ...

punctured at its own center, and could be any point therein. The symmedian point of a triangle with side lengths , and has homogeneous

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...

. An algebraic way to find the symmedian point is to express the triangle by three linear equations in two unknowns given by the

hesse normal form In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane \mathbb^2, a plane in Euclidean space \mathbb^3, or a hyperplane in higher dimensions.John Vince: ''Geometry for C ...

s of the corresponding lines. The solution of this

overdetermined system In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent equations, inconsistent (it has no solution) when constructed with random coeffi ...

found by the least squares method gives the coordinates of the point. It also solves the optimization problem to find the point with a minimal sum of squared distances from the sides. The

Gergonne point In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

of a triangle is the same as the symmedian point of the triangle's contact triangle.. The symmedian point of a triangle can be constructed in the following way: let the tangent lines of the circumcircle of through and meet at , and analogously define and ; then is the

tangential triangle In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertex (geometry), vertices. Thus ...

of , and the lines , and intersect at the symmedian point of . It can be shown that these three lines meet at a point using Brianchon's theorem. Line is a symmedian, as can be seen by drawing the circle with center through and . The French mathematician Émile Lemoine proved the existence of the symmedian point in 1873, and Ernst Wilhelm Grebe published a paper on it in 1847.

Simon Antoine Jean L'Huilier Simon Antoine Jean L'Huilier (or L'Huillier) (24 April 1750 in Geneva – 28 March 1840 in Geneva) was a Swiss mathematician of French Huguenot descent. He is known for his work in mathematical analysis and topology, and in particular the ...

had also noted the point in 1809.. For the extension to an irregular tetrahedron see symmedian.

Notes

References

External links

* {{mathworld, id=SymmedianPoint, title=Symmedian Point Triangle centers