Antiparallel

Two parallel lines form with a transversal equal (corresponding) angles, see the diagram below where pairs of corresponding angles are assigned the same number.

Two lines are said to be antiparallel with respect to a transversal if they still form equal angles with the latter but in an reversed order:

In triangle geometry, antiparallels make a frequent appearance when the transversal is taken to be one of the angles bisectors of a triangle, while the opposite side serves as one of the antiparallel lines. In this case, the other line is said to be antiparallel to that side. Orthic and tangential triangles have their sides antiparallel to the corresponding sides of the reference triangle.

The whole family of antiparallels admits an easy construction based on an property of cyclic quadrilaterals.

Let C(P) be a circle with center P passing through the vertices A and B of ΔABC. Assume it cuts BC (or its extension) in D and AC in E. Then DE is antiparallel to AB. In the case where both D and E are on the sides (as opposed to the extensions of the sides) of the triangle, quadrilateral ABDE is cyclic and convex. In this case, the opposite angles of ABDE add up to 180^o. Passing to the supplementary angles makes it obvious that DE is antiparallel to AB.

(In other two cases where one or both of D, E lie on the extensions of the sides one does not even have to consider the supplementary angles.)

Inscribed angles ADB and AEB are also either equal or supplementary underscoring once more the manner in which the antiparallels are constructed as chords of appropriate circles.

References

1. R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960, p. 172