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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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Special cases of Apollonius' problem」の詳細全文を読む スポンサード リンク
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