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Antiparallel lines

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated March 15, 2025
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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.

01Relations

  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.
Two lines and are 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.
Two lines and are 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.
Given two lines and , lines and are antiparallel with respect to and if .
Given two lines and , lines and are antiparallel with respect to and if .
In any quadrilateral inscribed in a circle, any two opposite sides are antiparallel with respect to the other two sides.
In any quadrilateral inscribed in a circle, any two opposite sides are antiparallel with respect to the other two sides.
red angles are of equal size, ED and the tangent in B are antiparallel to AC and are perpendicular to MB
red angles are of equal size, ED and the tangent in B are antiparallel to AC and are perpendicular to MB

02Conic sections

In an oblique cone, there are exactly two families of parallel planes whose sections with the cone are circles. One of these families is parallel to the fixed generating circle and the other is called by Apollonius the subcontrary sections.

If one looks at the triangles formed by the diameters of the circular sections (both families) and the vertex of the cone (triangles ABC and ADB), they are all similar. That is, if CB and BD are antiparallel with respect to lines AB and AC, then all sections of the cone parallel to either one of these circles will be circles. This is Book 1, Proposition 5 in Apollonius.

A cone with two directions of circular sections
A cone with two directions of circular sections
Side view of a cone with the two antiparallel directions of circular sections.
Side view of a cone with the two antiparallel directions of circular sections.
Triangles ABC and ADB are similar
Triangles ABC and ADB are similar
Watch videos about Antiparallel linesExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Antiparallel lines”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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