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Tucker Circles


TuckerCircles

The Tucker circles are a generalization of the cosine circle and first Lemoine circle which can be viewed as a family of circles obtained by parallel displacing sides of the corresponding cosine or Lemoine hexagon. No matter how the segments are displaced, the Tucker hexagon will close, and the 12 vertices will be concyclic. The cosine circle and first Lemoine circle correspond to the special case where three sides of the Tucker hexagon concur.

Let three equal lines P_1Q_1, P_2Q_2, and P_3Q_3 be drawn antiparallel to the sides of a triangle so that two (say P_2Q_2 and P_3Q_3) are on the same side of the third line as A_2P_2Q_3A_3. Then P_2Q_3P_3Q_2 is an isosceles trapezoid, i.e., P_3Q_2, P_1Q_3, and P_2Q_1 are parallel to the respective sides. The midpoints C_1, C_2, and C_3 of the antiparallels are on the respective symmedians and divide them proportionally. If T divides KO in the same ratio, TC_1, TC_2, TC_3 are parallel to the radii OA_1, OA_2, and OA_3 and equal. Since the antiparallels are perpendicular to the symmedians, they form equal chords of a circle, called a Tucker circle, which passes through the six given points and has center T on the line KO (Honsberger 1995, pp. 92-94).

Defining

 lambda=(KC_1)/(KA_1)=(KC_2)/(KA_2)=(KC_3)/(KA_3)=(KT)/(KO),
(1)

then the radius of the Tucker circle is

 R_T=Rsqrt(lambda^2+(1-lambda)^2tan^2omega),
(2)

where omega is the Brocard angle and R is the circumradius of the reference triangle (Johnson 1929, p. 274).

Tucker circles can also be parametrized by a parametric angle phi. The Tucker circle with parametric angle has radius

 R_phi=(sinomega)/(sin(omega+phi))R,
(3)

where omega is the Brocard angle and R the circumradius of the reference triangle (Gallatly 1913, p. 134), and trilinear center function

 alpha=cos(A-phi).
(4)

Special named Tucker circles are summarized in the following table, where r is the inradius, s the semiperimeter of the reference triangle, and S and S_omega is Conway triangle notation.

The Tucker circles are a coaxaloid system (Johnson 1929, p. 277).

The internal and external centers of similitudes of a Tucker circle with the nine-point circle lie on the Kiepert hyperbola (P. Moses, pers. comm., Jan. 3, 2005).

The two intersections of a Tucker circle with the Brocard axis have center functions

 alpha_+/-=ecos(A-phi)+/-cos(A+phi),
(5)

and the inner and outer centers of similitude with the Brocard circle are given by

 alpha_+/-=ecos(A-phi)+/-cos(A-phi),
(6)

where

 e^2=1-4sin^2omega
(7)

(pers. comm., P. Moses, Jan. 3, 2005).


See also

Antiparallel, Apollonius Circle, Brocard Angle, Circumcircle, Coaxaloid System, Cosine Circle, Cosine Hexagon, First Lemoine Circle, Gallatly Circle, Kenmotu Circle, Lemoine Hexagon, Parallelian, Taylor Circle, Tucker Hexagon

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References

Casey, J. "Lemoine's, Tucker's, and Taylor's Circle." Supp. Ch. §3 in 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., pp. 179-189, 1888.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 68, 1971.Gallatly, W. "Pivot Points. Tucker Circles." Ch. 12 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 109-119, 1913.Grinberg, D. and Yiu, P. "The Apollonius Circle as a Tucker Circle." Forum Geom. 2, 175-182, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200222index.html.Honsberger, R. "The Tucker Circles." Ch. 9 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 87-98, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 271-277 and 300-301, 1929.Lachlan, R. §133 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 77, 1893.Third, J. A. "Systems of circles analogous to Tucker circles." Proc. Edinburgh Math. Soc 17, 70-99, 1898.Yff, P. "A Generalization of the Tucker Circles." Forum Geom. 2, 71-87, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200210index.html.

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Tucker Circles

Cite this as:

Weisstein, Eric W. "Tucker Circles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TuckerCircles.html

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