Spieker Circle


The common incircle of the medial triangle DeltaM_AM_BM_C (left figure) and the congruent triangle DeltaQ_AQ_BQ_C, where Q_i are the midpoints of the line segment joining the Nagel point Na with the vertices of the original triangle DeltaABC (right figure).

The Spieker circle has circle function


which does not correspond to any named center. The center of the Spieker circle is called the Spieker center Sp, and the circle has radius


where r is the inradius and s is the semiperimeter of the reference triangle.

The Spieker circle passes through Kimberling centers X_i for i=3035, 3036, 3037, 3038, 3039, 3040, 3041, and 3042.

Spieker center table on floorSpieker center table against wall

The common area of Wolfram Research's front-end programming group contains a triangular table illustrating the construction of the Spieker circle. This table was built by Theodore Gray, director of user interfaces at Wolfram Research, using walnut and inlays of maple, the latter of which was obtained from a tree formerly standing in the yard of front end developer Chris Carlson. The triangular table DeltaABC has sides lengths (3, 4, 5), a Pythagorean triple. The larger inlaid circle is the incircle of DeltaABC, with the incenter I representing the point of concurrence of the triangle's angle bisectors. The smaller inlaid circle is the Spieker circle, which can be seen to correspond to the incircle of the medial triangle DeltaM_AM_BM_C. The triangle's cleavers are also shown, and concur in the Spieker center (which is therefore also the cleavance center). E. Pegg Jr. has posted a photo history of the construction of this table.

See also

Incircle, Medial Triangle, Midpoint, Nagel Point, Spieker Center

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Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 53, 1971.Honsberger, R. "The Nagel Point M and the Spieker Circle." §1.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 3-13, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 226-228, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Pegg, E. Jr. "Triangle Table History.", T. "Ein merkwürdiger Kreis um den Schwerpunkt des Perimeters des geradlinigen Dreiecks als Analogen des Kreises der neun Punkte." Archiv Math. u. Phys. 51, 10-14, 1870.

Referenced on Wolfram|Alpha

Spieker Circle

Cite this as:

Weisstein, Eric W. "Spieker Circle." From MathWorld--A Wolfram Web Resource.

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