Lemoine point

Lemoine point
Name	Lemoine point
Other names	Symmedian point, Grebe point
Type	Triangle center
Coordinates	trilinear: secA:secB:secC; barycentric: a^2:b^2:c^2
Discovered	19th century
Named for	Émile Lemoine

Lemoine point The Lemoine point is a triangle center defined as the concurrency point of the three symmedians of a triangle. It lies inside an acute triangle and is characterized by ratios involving the squares of side lengths; it is related to pedal circles, isogonal conjugation, and affine maps connecting classical centers.

Definition and basic properties

The Lemoine point is the common intersection of the three symmedians, each obtained from a corresponding median by isogonal reflection with respect to the angles at Euclid-era vertices such as Alexander the Great's namesake triangle or modern constructions by Émile Lemoine. It minimizes the sum of squared distances to the triangle sides in a weighted sense and is isogonal conjugate to the centroid of a given triangle; it also corresponds to the point whose trilinears are proportional to the secants of the angles, linking to centers like the circumcenter, incenter, orthocenter, Gergonne point, and Nagel point. The Lemoine point is fixed under the action of the triangle's symmetry group and participates in homographies that send the Brocard points and Brocard circle into other notable loci.

Construction methods

One classical construction uses reflections: reflect each median across the corresponding internal angle bisector to obtain three symmedians; their common intersection is the center. Alternatively, use cevians through vertices that cut the opposite sides in ratios proportional to the squares of adjacent sides—these cevians concur at the point. Projective constructions employ harmonic conjugates and the polar with respect to the circumcircle; inversion about the circumcircle transforms medians into symmedians, providing a construction via known centers like the nine-point center and the Euler line intersections. Practical compass-and-straightedge steps relate to drawing tangents to the circumcircle at the vertices, intersecting pairs to form the Lemoine triangle and deducing the point by intersections of corresponding cevians.

Trilinear and barycentric coordinates

In trilinear coordinates the Lemoine point is given by sec A : sec B : sec C, linking directly to the triangle's angles A, B, C as in classical analytic formulations used by August Ferdinand Möbius and Jean-Victor Poncelet. In barycentric coordinates it is proportional to a^2 : b^2 : c^2, where a, b, c denote side lengths opposite vertices A, B, C; this places the point in the registry of triangle centers described in tables by Émile Lemoine and later catalogues such as those by Clark Kimberling. These coordinates allow computation of distances to centers like the incenter, circumcenter, and centroid via barycentric metric relations and enable coordinate proofs of concurrency and collinearity with points like the Kosnita point and De Longchamps point.

Geometric relationships and symmetries

The Lemoine point is the isogonal conjugate of the centroid and lies on lines connecting vertices with corresponding symmedian feet, forming the Lemoine or symmedian triangle whose vertices are intersections of pairs of symmedians with opposite sides. It is the center of the Brocard circle in special alignments and participates in the Brocard axis together with the Brocard points and the circumcenter. The pedal triangle of the Lemoine point is homothetic to the medial triangle and exhibits symmetries under the triangle's dihedral group; affinities that send the triangle to an equilateral triangle map the Lemoine point to the common center of classical centers such as the Fermat point in degenerate limits. Moreover, the Lemoine point is the perspector of the triangle and its tangential triangle formed by tangents at the circumcircle, connecting it to the Gergonne triangle and contact triangle through perspectivity relations.

Special cases and notable triangles

In an equilateral triangle the Lemoine point coincides with the centroid, circumcenter, incenter, and orthocenter, collapsing many distinctions among centers. In isosceles triangles the Lemoine point lies on the axis of symmetry along with the Nagel point and Spieker center; in right triangles the position approaches the midpoint of the hypotenuse under limiting processes relating to the Thales circle. For triangles with integer sides studied by Pythagoras-inspired parametrizations or by modern enumerations like those of Smarandache and Erdős, barycentric coordinates a^2:b^2:c^2 give explicit rational relations when squares are commensurate, producing constructible loci tied to lattice-point problems studied by Gauss and Legendre.

Historical background and naming

The point is named after Émile Lemoine, who published on symmedians in the late 19th century and who contributed to triangle center theory alongside contemporaries such as Joseph-Louis Lagrange's successors and schoolmates in French geometry. Earlier notions related to symmedians appear in projective and synthetic work by Pascal-era and Desargues-inspired geometers; later formalization and tabulation of centers were advanced by Darij Grinberg-style cataloguers and systematic compilers like Clark Kimberling in the 20th century. The Lemoine point has since been featured in treatises by H. S. M. Coxeter, E. T. Whittaker, and various expositors in journals associated with institutions such as the École Polytechnique and the Académie des Sciences.

Category:Triangle centers