Antiparallel (mathematics)

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 $l_1$ 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 $l_1$ and $l_2$ are antiparallel with respect to the sides of an angle if they make the same angle $\mathrm{\angle }APC$ in the opposite senses with the bisector of that angle.
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 $\mathrm{\angle }1=\mathrm{\angle }2$.	In any quadrilateral inscribed in a circle, any two opposite sides are antiparallel with respect to the other two sides.

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

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