Antiparallel (mathematics)

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

Definitions

Given two lines $m_1 \,$ and $m_2 \,$, lines $l_1 \,$ and $l_2 \,$ are anti-parallel with respect to $m_1 \,$ and $m_2 \,$ if $\angle 1 = \angle 2 \,$ in Fig.1. If $l_1 \,$ and $l_2 \,$ are anti-parallel with respect to $m_1 \,$ and $m_2 \,$, then $m_1 \,$ and $m_2 \,$ are also anti-parallel with respect to $l_1 \,$ and $l_2 \,$.

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 $\angle APC$ in the opposite senses with the bisector of that angle (Fig.3).

Fig.1: Given two lines $m_1 \,$ and $m_2 \,$, lines $l_1 \,$ and $l_2 \,$ are anti-parallel with respect to $m_1 \,$ and $m_2 \,$ if $\angle 1 = \angle 2 \,$.

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 $l_1 \,$ and $l_2 \,$ are said to be antiparallel with respect to the sides of an angle if they make the same angle $\angle APC$ 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 $m_1 \,$ and $m_2 \,$ coincide, $l_1 \,$ and $l_2 \,$ 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

1. 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)
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.

References

1. 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

• A.B. Ivanov, Encyclopaedia of Mathematics - ISBN 1-4020-0609-8
• Weisstein, Eric W. "Antiparallel." From MathWorld--A Wolfram Web Resource.