TOPICS
Search

Symmedian Point


Symmedians

The point of concurrence K of the symmedians, sometimes also called the Lemoine point (in England and France) or the Grebe point (in Germany). Equivalently, the symmedian point is the isogonal conjugate of the triangle centroid G. In other words, let G be the triangle centroid of a triangle DeltaABC, AM_A, BM_B, and CM_C the medians of DeltaABC, AL_A, BL_B, and CL_C the angle bisectors of angles A, B, C, and AK_A, BK_B, and CK_C the reflections of AM_A, BM_B, and CM_C about AL_A, BL_B, and CL_C. Then K is the point of concurrence of the lines AK_A, BK_B, and CK_C. According to Honsberger (1995, p. 53), the symmedian point is "one of the crown jewels of modern geometry." The symmedian point is Kimberling center X_6.

The symmedian point has equivalent triangle center functions and

 alpha_6=a
(1)

(Honsberger 1995, p. 75), or

 alpha_6=sinA.
(2)

In exact trilinear coordinates, the symmedian point is the point for which alpha^2+beta^2+gamma^2 is a minimum (Honsberger 1995, pp. 75-76). A center X is the triangle centroid of its own pedal triangle iff it is the symmedian point. The symmedian point is the perspectivity center of a triangle and its tangential triangle.

The following table summarizes the symmedian points for named triangles that are Kimberling centers.

SymmedianPointDivision

In the above diagram with K the symmedian point,

 (AK)/(KK_A)=(b^2+c^2)/(a^2)
(3)

(Honsberger 1995, p. 76).

BrocardAxis

The symmedian point lies on the Brocard axis and the Fermat axis. It lies of the Brocard circle and is the center of the cosine circle. It also lies on the Jerabek hyperbola and the Thomson cubic.

Its distances from K to the sides of the triangle are

 KK_i=1/2a_itanomega,
(4)

where omega is the Brocard angle.

Distances to some other named triangle centers are given by

KG=1/(3(a^2+b^2+c^2))(sqrt(-a^6+3b^2a^4+3c^2a^4+3b^4a^2+3c^4a^2-15b^2c^2a^2-b^6-c^6+3b^2c^4+3b^4c^2))
(5)
KH=1/(4Delta(a^2+b^2+c^2))(sqrt(a^(10)-b^2a^8-c^2a^8+b^2c^2a^6-b^8a^2-c^8a^2+b^2c^6a^2+b^6c^2a^2+b^(10)+c^(10)-b^2c^8-b^8c^2))
(6)
KI=1/(a^2+b^2+c^2)sqrt(-1/((a+b+c))(abc(a^4-2ba^3-2ca^3+2b^2a^2+2c^2a^2+bca^2-2b^3a-2c^3a+bc^2a+b^2ca+b^4+c^4-2bc^3+2b^2c^2-2b^3c)))
(7)
KM=((a+b+c)^2IK)/(a^2-2ab+b^2-asc-2bc+c^2)
(8)
KO=(2abc(a^4-b^2a^2-c^2a^2+b^4+c^4-b^2c^2))/(4Delta(a^2+b^2+c^2)),
(9)

where G is the triangle centroid, H is the orthocenter, I is the incenter, M is the mittenpunkt, and O is the circumcenter.

BrocardCentroidLemoine

One Brocard line, triangle median, and symmedian (out of the three of each) are concurrent, with AOmega, CK, and BG meeting at a point, where Omega is the first Brocard point and G is the triangle centroid. Similarly, AOmega^', BG, and CK, where Omega^' is the second Brocard point, meet at a point which is the isogonal conjugate of the first (Johnson 1929, pp. 268-269).

SymmedianMidpoints

The line joining the midpoint of any side to the midpoint of the altitude on that side passes through K (left figure). In particular, the symmedian point of a right triangle is the midpoint of the altitude to the hypotenuse (right figure; Honsberger 1995, p. 59). The symmedian point K is the Steiner point of the first Brocard triangle.

SymmedianPointCircumcircle

Given a triangle DeltaABC, construct the triangle DeltaA^'B^'C^' obtained as the intersection of the lines extended from each vertex though the symmedian point K of DeltaABC with the circumcircle of DeltaABC. Then the symmedian point of DeltaA^'B^'C^' is again K (Honsberger 1995, p. 77).

The tangents to the circumcircle of a triangle at two of its vertices meet on the symmedian from the third vertex (Honsberger 1995, pp. 60-61). The Gergonne point of a triangle is the symmedian point of its contact triangle (Honsberger 1995, pp. 62-63). The symmedian point of a triangle is the triangle centroid of its pedal triangle. And finally, the lengths of the sides of the pedal triangle of the symmedian point are proportional to the lengths of the triangle medians of the original triangle (Honsberger 1995, p. 77).


See also

Angle Bisector, Brocard Angle, Brocard Axis, Brocard Diameter, Cosymmedian Triangles, First Lemoine Circle, Isogonal Conjugate, Lemoine Axis, Line at Infinity, Mittenpunkt, Pedal Triangle, Schoute Center, Steiner Points, Symmedian, Tangential Triangle, Triangle Centroid

Explore with Wolfram|Alpha

References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 170, 1888.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 65, 1971.Gallatly, W. "The Lemoine Point." §117 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 86, 1913.Honsberger, R. "The Symmedian Point." Ch. 7 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 53-77, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 217, 268-269, and 271-272, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Symmedian Point." http://faculty.evansville.edu/ck6/tcenters/class/sympt.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(6)=Symmedian Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X6.Mackay, J. S. "Early History of the Symmedian Point." Proc. Edinburgh Math. Soc. 11, 92-103, 1892-1893.Mackay, J. S. "Symmedians of a Triangle and Their Concomitant Circles." Proc. Edinburgh Math. Soc. 14, 37-103, 1896.

Referenced on Wolfram|Alpha

Symmedian Point

Cite this as:

Weisstein, Eric W. "Symmedian Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymmedianPoint.html

Subject classifications