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In geometry, anti-parallel lines can be defined with respect to either lines or angles.

Definitions

Given two lines $${\displaystyle m_{1}\,}$$ and $${\displaystyle m_{2}\,}$$, lines $${\displaystyle l_{1}\,}$$ and $${\displaystyle l_{2}\,}$$ are anti-parallel with respect to $${\displaystyle m_{1}\,}$$ and $${\displaystyle m_{2}\,}$$ if ∠ 1 = ∠ 2 {\displaystyle \angle 1=\angle 2\,} {\displaystyle \angle 1=\angle 2\,} in Fig.1. If $${\displaystyle l_{1}\,}$$ and $${\displaystyle l_{2}\,}$$ are anti-parallel with respect to $${\displaystyle m_{1}\,}$$ and $${\displaystyle m_{2}\,}$$ , then $${\displaystyle m_{1}\,}$$ and $${\displaystyle m_{2}\,}$$ are also anti-parallel with respect to $${\displaystyle l_{1}\,}$$ and $${\displaystyle l_{2}\,}$$ .

In any quadrilateral inscribed in a circle, any two opposite sides are anti-parallel with respect to the other two sides (Fig.2).

Two lines $$l_{1}$$ and $$l_{2}$$ are antiparallel with respect to the sides of an angle if and only if they make the same angle $${\displaystyle \angle APC}$$ in the opposite senses with the bisector of that angle (Fig.3).
Fig.1: Given two lines $${\displaystyle m_{1}\,}$$ and $${\displaystyle m_{2}\,}$$ , lines $${\displaystyle l_{1}\,}$$ and $${\displaystyle l_{2}\,}$$ are anti-parallel with respect to $${\displaystyle m_{1}\,}$$ and $${\displaystyle m_{2}\,}$$ if $${\displaystyle \angle 1=\angle 2\,}$$.

Fig.2: In any quadrilateral inscribed in a circle, any two opposite sides are anti-parallel with respect to the other two sides.

Fig.3: Two lines $${\displaystyle l_{1}\,}$$ and $${\displaystyle l_{2}\,}$$ are said to be antiparallel with respect to the sides of an angle if they make the same angle $${\displaystyle \angle APC}$$ in the opposite senses with the bisector of that angle. Notice that our previous angles 1 and 2 are still equivalent.

Fig.4: If the lines $${\displaystyle m_{1}\,}$$ and $${\displaystyle m_{2}\,}$$ coincide, $${\displaystyle l_{1}\,}$$ and $${\displaystyle l_{2}\,}$$ are said to be anti-parallel with respect to a straight line.

Antiparallel vectors

In a Euclidean space, two directed line segments, often called vectors in applied mathematics, are antiparallel, if they are supported by parallel lines and have opposite directions.[1] In that case, one of the associated Euclidean vectors is the product of the other by a negative number.
Relations

The line joining the feet to two altitudes of a triangle is antiparallel to the third side.(any cevians which 'see' the third side with the same angle create antiparallel lines)
The tangent to a triangle's circumcircle at a vertex is antiparallel to the opposite side.
The radius of the circumcircle at a vertex is perpendicular to all lines antiparallel to the opposite sides.

References

Harris, John; Harris, John W.; Stöcker, Horst (1998). Handbook of mathematics and computational science. Birkhäuser. p. 332. ISBN 0-387-94746-9., Chapter 6, p. 332

Sources

A.B. Ivanov, Encyclopaedia of Mathematics - ISBN 1-4020-0609-8
Weisstein, Eric W. "Antiparallel." From Mathworld—A Wolfram Web Resource. [1]