Antiparallel (mathematics)
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In geometry, anti-parallel lines can be defined with respect to either lines or angles.
[edit] Definitions
Given two lines 
and 
, lines 
and 
are anti-parallel with respect to and if 
.
and , lines and are anti-parallel with respect to and if .If and and are anti-parallel with respect to and , then and and are also anti-parallel with respect to and .
In any quadrilateral inscribed in a circle, any two opposite sides are anti-parallel with respect to the other two sides.
Two lines and are said to be antiparallel with respect to the sides of an angle if they make the same angle 
in the opposite senses with the bisector of that angle.
and are said to be antiparallel with respect to the sides of an angle if they make the same angle in the opposite senses with the bisector of that angle. Notice that our previous angles 1 and 2 are still equivalent.
and coincide, and are said to be anti-parallel with respect to a straight line.[edit] 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.
[edit] References
- A.B. Ivanov, Encyclopaedia of Mathematics - ISBN 1402006098
- Weisstein, Eric W. "Antiparallel." From MathWorld--A Wolfram Web Resource. [1]
