Special Cases of Apollonius' Problem

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:

  1. three points (denoted PPP, generally 1 solution)
  2. three lines (denoted LLL, generally 4 solutions)
  3. one line and two points (denoted LPP, generally 2 solutions)
  4. two lines and a point (denoted LLP, generally 2 solutions)
  5. one circle and two points (denoted CPP, generally 2 solutions)
  6. one circle, one line, and a point (denoted CLP, generally 4 solutions)
  7. two circles and a point (denoted CCP, generally 4 solutions)
  8. one circle and two lines (denoted CLL, generally 8 solutions)
  9. 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.

Read more about Special Cases Of Apollonius' Problem:  Historical Introduction, Rules of Euclidean Constructions, Preliminary Results, Special Cases With No Solutions, See Also

Famous quotes containing the words special, cases and/or problem:

    “He’s leaving Germany by special request of the Nazi government. First he sends a dispatch about Danzig and how 10,000 German tourists are pouring into the city every day with butterfly nets in their hands and submachine guns in their knapsacks. They warn him right then. What does he do next? Goes to a reception at von Ribbentropf’s and keeps yelling for gefilte fish!”
    —Billy Wilder (b. 1906)

    “You all know that even when women have full rights, they still remain fatally downtrodden because all housework is left to them. In most cases housework is the most unproductive, the most barbarous and the most arduous work a woman can do. It is exceptionally petty and does not include anything that would in any way promote the development of the woman.”
    —Vladimir Ilyich Lenin (1870–1924)

    “Hypocrisy is the essence of snobbery, but all snobbery is about the problem of belonging.”
    —Alexander Theroux (b. 1940)