Special Cases of Apollonius' Problem - Preliminary Results

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.

• A circle is tangent to a point if it passes through the point, and tangent to a line if they intersect at a single point P or if the line is perpendicular to a radius drawn from the circle's center to P.
• Circles tangent to two given points must lie on the perpendicular bisector.
• Circles tangent to two given lines must lie on the angle bisector.
• Tangent line to a circle from a given point draw semicircle centered on the midpoint between the center of the circle and the given point.
• Power of a point and the harmonic mean
• The radical axis of two circles is the set of points of equal tangents, or more generally, equal power.
• Circles may be inverted into lines and circles into circles.
• If two circles are internally tangent, they remain so if their radii are increased or decreased by the same amount. Conversely, if two circles are externally tangent, they remain so if their radii are changed by the same amount in opposite directions, one increasing and the other decreasing.