Taylor Circle


From the feet H_A, H_B, and H_C of each altitude of a triangle DeltaABC, draw lines (H_AP_A,H_AQ_A), (H_BP_B,H_BQ_B), (H_CP_C,H_CQ_C) perpendicular to the adjacent sides, as illustrated above. Then the points P_A, P_B, P_C, Q_A, Q_B, and Q_C are concyclic, and the circle passing through these points is called the Taylor circle. Here, P_AQ_A, P_BQ_B, and P_CQ_C are antiparallel to the sides BC, CA, and AB, respectively.

Furthermore, the figures AH_CHH_B and AP_AH_AQ_A are similar, where H is the orthocenter of DeltaABC, P_AQ_A is parallel to H_BH_C, and P_AQ_A bisects H_AH_B and H_AH_C.

If R is the circumradius of the reference triangle, then


Also, if the triangle is acute, this is equal to


(Johnson 1929, p. 277).

The Taylor circle has circle function


which corresponds to Kimberling center X_(394). The center of the circle has trilinear function


which is Kimberling center X_(389), is the center of the Spieker circle of the orthic triangle of the reference triangle (Johnson 1929, p. 277), and is called the Taylor center.

The radius of the Taylor circle is given by


No notable triangle centers lie on the Taylor circle.

The Taylor circle is a Tucker circle with parameter


There are a number of remarkable properties satisfied by the figure obtained in the construction of the Taylor circle:

1. The feet of the perpendiculars from a given altitude foot are concyclic with the opposite vertex.

2. The two feet of the perpendiculars which are closest to a given vertex are concyclic with the feet of the altitudes on the corresponding sides.

3. The two feet of the perpendiculars which are closest to a give vertex are concyclic with that vertex and with the intersection of the perpendiculars.

4. The three circles through the orthocenter and the feet of the perpendiculars on a given side intersect pairwise along the altitudes.

The first three of these follow from that fact that ∠PQR=90 degrees is equivalent to Q lying on the circle with diameter PR, and the fourth follows from the concurrence of the pairwise radical axes of three circles (the three circles being two of the circles through the orthocenter and the feet of the perpendiculars on a given side, and the Taylor circle).

See also

Taylor Center, Tucker Circles

Portions of this entry contributed by Darij Grinberg

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Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952.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, pp. 71-73, 1971.Gallatly, W. "The Taylor Circle." §165 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 118-119, 1913.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 277, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 78, 1893.Taylor, H. M. Proc. London Math. Soc. 15.

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Taylor Circle

Cite this as:

Grinberg, Darij and Weisstein, Eric W. "Taylor Circle." From MathWorld--A Wolfram Web Resource.

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