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In 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 ...
such that the two products of opposite side lengths are equal. That is,
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 . 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 and arbitrarily on the circle, letting be any point exterior to the circle on line , and setting and 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 . Then is an Apollonius quadrilateral.
If , , and are fixed, then the locus of points that form an Apollonius quadrilateral is the set of points where the ratio of distances to and , , is the fixed ratio ; 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 having a fixed ratio of distances to two given points and , 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 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 ...
of , and the resulting quadrilateral is a kite.
See also
*Tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex polygon, convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This cir ...
, where sums rather than products of opposite sides are equal
References
{{Polygons Types of quadrilaterals Greek mathematics