In geometry, an Apollonius quadrilateral is a quadrilateral formula_1 such that the two products of opposite side lengths are equal. That is,
formula_2
An equivalent way of stating this definition is that the cross ratio of the four points is formula_3. It is allowed for the quadrilateral sides to cross.
The Apollonius quadrilaterals are important in inversive geometry, because the property of being an Apollonius quadrilateral is preserved by Möbius transformations, and every continuous transformation of the plane that preserves all Apollonius quadrilaterals must be a Möbius transformation.
Every kite is an Apollonius quadrilateral. A special case of the Apollonius quadrilaterals are the harmonic quadrilaterals; these are cyclic Apollonius quadrilaterals, inscribed in a given circle. They may be constructed by choosing two opposite vertices formula_4 and formula_5 arbitrarily on the circle, letting formula_6 be any point exterior to the circle on line formula_7, and setting formula_8 and formula_9 to be the two points where the circle is touched by the tangent lines to circles through formula_6. Then formula_1 is an Apollonius quadrilateral.
If formula_4, formula_8, and formula_5 are fixed, then the locus of points formula_9 that form an Apollonius quadrilateral formula_1 is the set of points where the ratio of distances to formula_4 and formula_5, formula_19, is the fixed ratio formula_20; this is just a rewritten form of the defining equation for an Apollonius quadrilateral. As Apollonius of Perga proved, the set of points formula_9 having a fixed ratio of distances to two given points formula_4 and formula_5, and therefore the locus of points that form an Apollonius quadrilateral, is a circle in a family of circles called the Apollonian circles. Because formula_8 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 of formula_7, and the resulting quadrilateral is a kite.