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In
Tangencies: Circular Angle Bisectors
The Geometry Junkyard, David Eppstein, 1999. If the two circles ''α'' and ''β'' cross each other, then their two circles of antisimilitude each pass through both crossing points, and bisect the angles formed by the arcs of ''α'' and ''β'' as they cross. If a circle ''γ'' crosses circles ''α'' and ''β'' at equal angles, then ''γ'' is crossed orthogonally by one of the circles of antisimilitude of ''α'' and ''β''; if ''γ'' crosses ''α'' and ''β'' in
In inversive geometry
Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotions from any exterior source. An inversive heat source would be a heat source where all t ...
, the circle of antisimilitude (also known as mid-circle) of two circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s, ''α'' and ''β'', is a reference circle for which ''α'' and ''β'' are inverses of each other. If ''α'' and ''β'' are non-intersecting or tangent, a single circle of antisimilitude exists; if ''α'' and ''β'' intersect at two points, there are two circles of antisimilitude. When ''α'' and ''β'' are congruent, the circle of antisimilitude degenerates to a line of symmetry through which ''α'' and ''β'' are reflections of each other...
Properties
If the two circles ''α'' and ''β'' cross each other, another two circles ''γ'' and ''δ'' are each tangent to both ''α'' and ''β'', and in addition ''γ'' and ''δ'' are tangent to each other, then the point of tangency between ''γ'' and ''δ'' necessarily lies on one of the two circles of antisimilitude. If ''α'' and ''β'' are disjoint and non-concentric, then the locus of points of tangency of ''γ'' and ''δ'' again forms two circles, but only one of these is the (unique) circle of antisimilitude. If ''α'' and ''β'' are tangent or concentric, then the locus of points of tangency degenerates to a single circle, which again is the circle of antisimilitude.The Geometry Junkyard, David Eppstein, 1999. If the two circles ''α'' and ''β'' cross each other, then their two circles of antisimilitude each pass through both crossing points, and bisect the angles formed by the arcs of ''α'' and ''β'' as they cross. If a circle ''γ'' crosses circles ''α'' and ''β'' at equal angles, then ''γ'' is crossed orthogonally by one of the circles of antisimilitude of ''α'' and ''β''; if ''γ'' crosses ''α'' and ''β'' in
supplementary angles
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
, it is crossed orthogonally by the other circle of antisimilitude, and if ''γ'' is orthogonal to both ''α'' and ''β'' then it is also orthogonal to both circles of antisimilitude.
For three circles
Suppose that, for three circles ''α'', ''β'', and ''γ'', there is a circle of antisimilitude for the pair (''α'',''β'') that crosses a second circle of antisimilitude for the pair (''β'',''γ''). Then there is a third circle of antisimiltude for the third pair (''α'',''γ'') such that the three circles of antisimilitude cross each other in two triple intersection points. Altogether, at most eight triple crossing points may be generated in this way, for there are two ways of choosing each of the first two circles and two points where the two chosen circles cross. These eight or fewer triple crossing points are the centers of inversions that take all three circles ''α'', ''β'', and ''γ'' to become equal circles. For three circles that are mutually externally tangent, the (unique) circles of antisimilitude for each pair again cross each other at 120° angles in two triple intersection points that are theisodynamic point
In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the dis ...
s of the triangle formed by the three points of tangency.
See also
*Inversive geometry
Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotions from any exterior source. An inversive heat source would be a heat source where all t ...
* Limiting point (geometry), the center of an inversion that transforms two circles into concentric position
*Radical axis
In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. ...
References
External links
*{{mathworld, title=Midcircle, id=Midcircle Circles Inversive geometry