geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, two lines

l_1

and

l_2

are antiparallel with respect to a given line

m

if they each make

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

s with

m

in opposite

senses A sense is a biological system used by an organism for sensation, the process of gathering information about the surroundings through the detection of stimuli. Although, in some cultures, five human senses were traditionally identified as su ...

. 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 In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''se ...

m_1

and

m_2.

In any

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

, any two opposite sides are antiparallel with respect to the other two sides.

Relations

# The line joining the feet to two

altitudes Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometry, geographical s ...

of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

is antiparallel to the third side. (any

cevian In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. Medians and angle bisectors are special cases of cevians. The name ''cevian'' comes from the Italian mathematician Giov ...

s which 'see' the third side with the same angle create antiparallel lines) # The tangent to a triangle's

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

at a vertex is antiparallel to the opposite side. # The radius of the circumcircle at a vertex is

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

to all lines antiparallel to the opposite sides. Antiparallel_aussagen

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. 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 Apollonius () is a masculine given name which may refer to: People Ancient world Artists * Apollonius of Athens (sculptor) (fl. 1st century BC) * Apollonius of Tralles (fl. 2nd century BC), sculptor * Apollonius (satyr sculptor) * Apo ...

References

* *A.B. Ivanov
''Anti-parallel straight lines''
In: Encyclopaedia of Mathematics - *

External links

* {{cci Elementary geometry