Antiparallel lines

{{for|vectors pointed in opposite directions|Antiparallel vectors}}

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.

style="margin: 1em auto 1em auto;"

valign=top

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.

File:Antiparallel_aussagen.svg

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 fixed

generating circle and the other is called by Apollonius the subcontrary sections.{{cite book

| last = Heath

| first = Thomas Little

| author-link = Thomas Heath (classicist)

| date = 1896

| title = Treatise on conic sections

| url = https://archive.org/details/treatiseonconics00apolrich/page/2/mode/2up

| page = 2

| publisher = Cambridge, University press

}}

style="margin: 1em auto 1em auto;"

valign=top

|File:Apollonius-5 2.png

|File:Apollonius-5 1.png

File:Apollonius 6-6 13.png

If one looks at the triangles formed by the diameters of the circular sections (both families) and the vertex of the cone (triangles {{mvar|ABC}} and {{mvar|ADB}}), they are all similar. That is, if {{mvar|CB}} and {{mvar|BD}} are antiparallel with respect to lines {{mvar|AB}} and {{mvar|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.

References

{{cite journal |last1=Blaga |first1=Cristina |last2=Blaga |first2=Paul A. |year=2018 |title=Directed Angles |journal=Didactica Mathematica |volume=36 |pages=25–40 |url=https://www.math.ubbcluj.ro/~didactica/pdfs/2018/didmath2018-03.pdf }}
A.B. Ivanov: [https://encyclopediaofmath.org/index.php?title=Anti-parallel_straight_lines&oldid=13704 Anti-parallel straight lines]. In: Encyclopaedia of Mathematics - {{ISBN|1-4020-0609-8}}
{{MathWorld|title=Antiparallel|urlname=Antiparallel}}

External links

{{cci}}

Category:Elementary geometry