and are said to be antiparallel with respect
to the sides of an angle if they make the same angle in the opposite senses with the
bisector of that angle. If and are antiparallel with respect to and , then the latter are also antiparallel with respect to the
former. Furthermore, if
and are antiparallel, then the points , , ,
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 , a symmedian bisects all segments antiparallel to a given side (Honsberger 1995, p. 88). Furthermore, every antiparallel
to in is parallel to the tangent
to the circumcircle of at (Honsberger 1995, p. 98).