Antiparallel lines explained
and
are
antiparallel with respect to a given line
if they each make
congruent angles with
in opposite
senses. More generally, lines
and
are
antiparallel with respect to another pair of lines
and
if they are antiparallel with respect to the angle bisector of
and
In any cyclic quadrilateral, any two opposite sides are antiparallel with respect to the other two sides.
Relations
- 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)
- The tangent to a triangle's circumcircle at a vertex is antiparallel to the opposite side.
- The radius of the circumcircle at a vertex is perpendicular to all lines antiparallel to the opposite sides.
Conic 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 fixedgenerating circle and the other is called by Apollonius the subcontrary sections.[1]
If one looks at the triangles formed by the diameters of the circular sections (both families) and the vertex of the cone (triangles and), they are all similar. That is, if and are
antiparallel with respect to lines and, then all sections of the cone parallel to either one of these circles will be circles. This is Book 1, Proposition 5 in Apollonius.
References
Notes and References
- Book: Heath
, Thomas Little
. Thomas Heath (classicist) . 1896 . Treatise on conic sections . 2.
