In Euclidean geometry, Apollonius' problem is to construct all the circles that are tangent to three given circles. Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to:
three points (denoted PPP, generally 1 solution) three lines (denoted LLL, generally 4 solutions) one line and two points (denoted LPP, generally 2 solutions) two lines and a point (denoted LLP, generally 2 solutions) one circle and two points (denoted CPP, generally 2 solutions) one circle, one line, and a point (denoted CLP, generally 4 solutions) two circles and a point (denoted CCP, generally 4 solutions) one circle and two lines (denoted CLL, generally 8 solutions) two circles and a line (denoted CCL, generally 8 solutions)
In a different type of limiting case, the three given geometrical elements may have a special arrangement, such as constructing a circle tangent to two parallel lines and one circle.
Historical introduction
Like most branches of mathematics, Euclidean geometry is concerned with proofs of general truths from a minimum of postulates. For example, a simple proof would show that at least two angles of an isosceles triangle are equal. One important type of proof in Euclidean geometry is to show that a geometrical object can be constructed with a compass and an unmarked straightedge; an object can be constructed if and only if (iff) (something about no higher than square roots are taken). Therefore, it is important to determine whether an object can be constructed with compass and straightedge and, if so, how it may be constructed.
Euclid developed numerous constructions with compass and straightedge. Examples include: regular polygons such as the pentagon and hexagon, a line parallel to another that passes through a given point, etc. Many rose windows in Gothic Cathedrals, as well as some Celtic knots, can be designed using only Euclidean constructions. However, some geometrical constructions are not possible with those tools, including the heptagon and trisecting an angle.
Apollonius contributed many constructions, namely, finding the circles that are tangent to three geometrical elements simultaneously, where the "elements" may be a point, line or circle.
Rules of Euclidean constructions
In Euclidean constructions, five operations are allowed:
Draw a line through two points
Draw a circle through a point with a given center
Find the intersection point of two lines
Find the intersection points of two circles
Find the intersection points of a line and a circle
The initial elements in a geometric construction are called the "givens", such as a given point, a given line or a given circle.
Example 1: Perpendicular bisector
To construct the perpendicular bisector of the line segment between two points requires two circles, each centered on an endpoint and passing through the other endpoint (operation 2). The intersection points of these two circles (operation 4) are equidistant from the endpoints. The line through them (operation 1) is the perpendicular bisector.
Example 2: Angle bisector
To generate the line that bisects the angle between two given rays requires a circle of arbitrary radius centered on the intersection point P of the two lines (2). The intersection points of this circle with the two given lines (5) are T1 and T2. Two circles of the same radius, centered on T1 and T2, intersect at points P and Q. The line through P and Q (1) is an angle bisector. Rays have one angle bisector; lines have two, perpendicular to one another.
Preliminary results
A few basic results are helpful in solving special cases of Apollonius' problem. Note that a line and a point can be thought of as circles of infinitely large and infinitely small radius, respectively.
