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 $angle\; 1\; =\; angle\; 2\; ,$.

If $l\_1\; ,$ and $l\_2\; ,$ and are anti-parallel with respect to $m\_1\; ,$ and $m\_2\; ,$, then $m\_1\; ,$ and $m\_2\; ,$ and 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.

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.

Relations

# The line joining the feet to two altitudes of a triangle is antiparallel to the third side.
# 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

*A.B. Ivanov, Encyclopaedia of Mathematics - ISBN 1402006098
*Weisstein, Eric W. "Antiparallel." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/Antiparallel.html]