アポロニウスの円 : "中心と半径"か"直径の両端"かの考察を通して

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"The Locus of Apollonius" is old and well known, and it has been playing an important part in the study of locus. Accordingly, it has been dealt with in teaching materials in high school mathematics. The solution of this locus in general is methodically settled by means of either elementary or analytic geometry. In the case of the latter, it is usual to search for the center and the radius when finding the equation of the locus. In some high school mathematics textbooks there is a "Note" as follows: Note. Let A and B be two fixed points in the plane and let m and n be given positive constants. Then the locus of a point P, such that AP:BP=m:n is either (i) (if m≠n) a circle with a diameter where both ends divide internally or externally line segment AB into the ratio of m to n, (this circle is called the "Circle of Apollonius"), or (ii) (if m=n) the perpendicular bisector of line segment AB. However, elementary geometry is not being taught in many high schools in Japan, so it is almost impossible to find the equation by using the Note mentioned above. In this article, we have examined the answer using analytic geometry (rectangular coordinates, position vectors, complex numbers, and polar coordinates), so that we have not become deeply involved in solving the problem with elementary geometry. We especially analyzed "the center and the radius" and "both ends of the diameter" in comparison to each other. As a result of our examination, we concluded that "both ends of the diameter" is better than "the center and the radius" for finding the equation. From the viewpoint of math-education, we feel that the locus of Apollonius remains a very useful tool.

アポロニウスの円 : "中心と半径"か"直径の両端"かの考察を通して

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