geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (

medians The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...

reflected at the associated

angle bisectors 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'' (a line that passes throug ...

) of a triangle.

Ross Honsberger Ross Honsberger (1929–2016) was a Canadian mathematician and author on recreational mathematics. Life Honsberger studied mathematics at the University of Toronto, with a bachelor's degree, and then worked for ten years as a teacher in Toronto ...

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. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the ...

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 o ...

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 t ...

. 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 The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in \mathbb^2 or a plane in Euclidean space \mathbb^3 or a hyperplane in higher dimensions.John Vince: ''Geometry for Computer Graphics''. ...

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 (it has no solution) when constructed with random coefficients. However, an over ...

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 vertices. Thus the incircle of the ...

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 In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1 ...

. Line is a symmedian, as can be seen by drawing the circle with center through and . The French mathematician

Émile Lemoine Émile Michel Hyacinthe Lemoine (; 22 November 1840 – 21 February 1912) 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, mo ...

proved the existence of the symmedian point in 1873, and

Ernst Wilhelm Grebe Ernst is both a surname and a given name, the German, Dutch, and Scandinavian form of Ernest. Notable people with the name include: Surname * Adolf Ernst (1832–1899) German botanist known by the author abbreviation "Ernst" * Anton Ernst (1975-) ...

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 gen ...

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