Antiparallel (mathematics)

Last updated August 28, 2023

In geometry, two lines $l_{1}$ and $l_{2}$ are antiparallel with respect to a given line $m$ if they each make congruent angles with $m$ in opposite senses. More generally, lines $l_{1}$ and $l_{2}$ are antiparallel with respect to another pair of lines $m_{1}$ and $m_{2}$ if they are antiparallel with respect to the angle bisector of $m_{1}$ and $m_{2}.$

In any cyclic quadrilateral, any two opposite sides are antiparallel with respect to the other two sides.

Lines
l
1
{\displaystyle l_{1}}
and
l
2
{\displaystyle l_{2}}
are antiparallel with respect to the line
m
{\displaystyle m}
if they make the same angle with
m
{\displaystyle m}
in the opposite senses. Anti3.svg — Lines $l_{1}$ and $l_{2}$ are *antiparallel* with respect to the line $m$ if they make the same angle with $m$ in the opposite senses.

Two lines
l
1
{\displaystyle l_{1}}
and
l
2
{\displaystyle l_{2}}
are antiparallel with respect to the sides of an angle if they make the same angle
[?]
A
P
C
{\displaystyle \angle APC}
in the opposite senses with the bisector of that angle. Anti2.svg — Two lines $l_{1}$ and $l_{2}$ are *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.

Given two lines
m
1
{\displaystyle m_{1}}
and
m
2
{\displaystyle m_{2}}
, lines
l
1
{\displaystyle l_{1}}
and
l
2
{\displaystyle l_{2}}
are antiparallel with respect to
m
1
{\displaystyle m_{1}}
and
m
2
{\displaystyle m_{2}}
if
[?]
1
=
[?]
2
{\displaystyle \angle 1=\angle 2}
. Anti1.svg — Given two lines $m_{1}$ and $m_{2}$ , lines $l_{1}$ and $l_{2}$ are antiparallel with respect to $m_{1}$ and $m_{2}$ if $\angle 1=\angle 2$ .

In any quadrilateral inscribed in a circle, any two opposite sides are antiparallel with respect to the other two sides. Anti5.svg — In any quadrilateral inscribed in a circle, any two opposite sides are antiparallel with respect to the other two sides.

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.

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References

Blaga, Cristina; Blaga, Paul A. (2018). "Directed Angles" (PDF). Didactica Mathematica. 36: 25–40.
A.B. Ivanov, Encyclopaedia of Mathematics - ISBN 1-4020-0609-8
Weisstein, Eric W. "Antiparallel." From MathWorld—A Wolfram Web Resource.

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