Equable shape

Template:Too technical 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 $\mathrm{\angle }1=\mathrm{\angle }2$.

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 $\mathrm{\angle }1=\mathrm{\angle }2$. If $l_1$ and $l_2$ are anti-parallel with respect to $m_1$ and $m_2$, then $m_1$ and $m_2$ 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 $\mathrm{\angle }APC$ 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 $\mathrm{\angle }APC$ 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 $ℝ$ (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.(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

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