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アポロニウスの問題

出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2023/01/10 01:20 UTC 版)

ユークリッド平面幾何学においてアポロニウスの問題（英: Problem of Apollonius）とは、平面において与えられた3つの円に接する円を描く問題である(図 1)。ペルガのアポロニウス (ca. 262 BC – ca. 190 BC)が彼の著作「接触」 Ἐπαφαί (Epaphaí, "Tangencies")においてこの有名な問題を提起し、解決した。この著作「接触」は現在失われているが、アレキサンドリアのパップスによる、アポロニウスの成果がまとめられた4世紀のレポートは現存している。3つの与円^{[注釈 1]}は一般的に、その3つの円に接する8つの相異なる円を持ち(図 2)、この円が3つの円を内部に持つか外部に持つかはそれぞれ異なる。すなわち、それぞれの円は、与えられた3つの円のうち一部を内部に持ち（残りの円は外部に持つ）、濃度が3の集合の部分集合は 2³ = 8 つ存在するため、そのような円は8つ存在する。

注釈

^ 本項では与えられた円（Given circles）をニアーイェシュ H.「アポロニウスの問題の画法幾何学的解法」『図学研究』第25巻第1号、日本図学会、1991年、 35-37頁。に従って与円と訳す。

参考文献

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