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 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 and , lines and are anti-parallel with respect to and if .
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.
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 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.
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 and coincide, and are said to be anti-parallel with respect to a straight line.
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.

[edit] 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.

[edit] References

  • A.B. Ivanov, Encyclopaedia of Mathematics - ISBN 1402006098
  • Weisstein, Eric W. "Antiparallel." From MathWorld--A Wolfram Web Resource. [1]