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 ...

, an Apollonius quadrilateral is a

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

ABCD

such that the two products of opposite side lengths are equal. That is,

\overline\cdot\overline=\overline\cdot\overline.

An equivalent way of stating this definition is that the

cross ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defin ...

of the four points is

\pm 1

. It is allowed for the quadrilateral sides to cross. The Apollonius quadrilaterals are important in

inversive geometry In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry ...

, because the property of being an Apollonius quadrilateral is preserved by

Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...

s, and every continuous transformation of the plane that preserves all Apollonius quadrilaterals must be a Möbius transformation. Every

kite A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...

is an Apollonius quadrilateral. A special case of the Apollonius quadrilaterals are the

harmonic quadrilateral In Euclidean geometry, a harmonic quadrilateral is a quadrilateral whose four vertices lie on a circle, and whose pairs of opposite edges have equal products of lengths. Harmonic quadrilaterals have also been called harmonic quadrangles. They ar ...

s; these are cyclic Apollonius quadrilaterals, inscribed in a given

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

. They may be constructed by choosing two opposite vertices

A

and

C

arbitrarily on the circle, letting

E

be any point exterior to the circle on line

AC

, and setting

B

and

D

to be the two points where the circle is touched by the

tangent lines to circles In Euclidean geometry, Euclidean plane geometry, a tangent line to a circle is a Line (geometry), line that touches the circle at exactly one Point (geometry), point, never entering the circle's interior. Tangent lines to circles form the subject ...

through

E

. Then

ABCD

is an Apollonius quadrilateral. If

A

B

, and

C

are fixed, then the locus of points

D

that form an Apollonius quadrilateral

ABCD

is the set of points where the ratio of distances to

A

and

C

\overline/\overline

, is the fixed ratio

\overline/\overline

; this is just a rewritten form of the defining equation for an Apollonius quadrilateral. As

Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...

proved, the set of points

D

having a fixed ratio of distances to two given points

A

and

C

, and therefore the locus of points that form an Apollonius quadrilateral, is a circle in a family of circles called the

Apollonian circles In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. T ...

. Because

B

defines the same ratio of distances, it lies on the same circle. In the case where the fixed ratio is one, the circle degenerates to a line, the

perpendicular 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 ...

AC

, and the resulting quadrilateral is a kite.

References

{{Polygons Types of quadrilaterals Greek mathematics

See also

References