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

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.

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.

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.

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 vector space over $\mathbb{R}$ (or some other ordered field), two nonzero vectors are called antiparallel if they are parallel but have opposite directions. In that case, one is a negative scalar times the other.

Relations

1. The line joining the feet to two altitudes of a triangle is antiparallel to the third side.
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 1402006098
• Weisstein, Eric W. "Antiparallel." From MathWorld--A Wolfram Web Resource. [1]