Two lines PQ and RS are said to be antiparallel with respect to the sides of an angle A if they make the same angle in the opposite senses with the bisector of that angle. If PQ and RS are antiparallel with respect to PR and QS, then the latter are also antiparallel with respect to the former. Furthermore, if PQ and RS are antiparallel, then the points P, Q, R, and S are concyclic (Johnson 1929, p. 172; Honsberger 1995, pp. 87-88).

There are a number of fundamental relationships involving a triangle and antiparallel lines (Johnson 1929, pp. 172-173).

1. The line joining the feet to two altitudes of a triangle is antiparallel to the third side.

2. The tangent to a triangle's circumcircle at a vertex is antiparallel to the opposite side.

3. The radius of the circumcircle at a vertex is perpendicular to all lines antiparallel to the opposite sides.


In a triangle DeltaABC, a symmedian BK bisects all segments antiparallel to a given side AC (Honsberger 1995, p. 88). Furthermore, every antiparallel to BC in DeltaABC is parallel to the tangent to the circumcircle of DeltaABC at A (Honsberger 1995, p. 98).

See also

Angle, Concyclic, Cosine Circle, Cosine Hexagon, First Lemoine Circle, Hyperparallel, Lemoine Hexagon, Parallel, Tucker Circles, Tucker Hexagon

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Casey, J. "Theory of Isogonal and Isotomic Points, and of Antiparallel and Symmedian Lines." Supp. Ch. §1 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. 165-173, 1888.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 65, 1971.Honsberger, R. "Parallels and Antiparallels." §9.1 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 87-88, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 172, 1929.Lachlan, R. §113 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 63, 1893.Phillips, A. W. and Fisher, I. Elements of Geometry. New York: American Book Co., 1896.

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Cite this as:

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

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