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]

Graduate Studies in Mathematics

Hellenica World - Scientific Library

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