The Circle of Apollonius (II) : To decide "the center and the radius" of the circle using elementary geometry

Abstract

We have argued about the formalization of the locus of Apollonius using analytic geometry in the articles). It can be expressed in two ways as follows: one is done by means of finding out "a center and a radius" of the circle determined by the locus of Apollonius; the other by finding out "both ends of the diameter" of its circle. We presented three typical methods on determining "both ends of the diameter" of the circle in the elementary geometrical style. Besides, we thought over the significance of teaching the formalization of the locus of Apollonius in senior high school. It is usual, when dealing with this locus in senior high school, to decide "the center and the radius" of the circle using analytic geometry. But it is always the case to find out "both ends of the diameter" of the circle using elementary geometry. We scarcely find how to decide "the center and the radius" of the circle using elementary geometry. Can it be possible to treat the method in senior high school teaching? This question motivated us to write this article. This article aims at making clear the followings: (1) to construct any point satisfying the condition of the locus of Apollonius in the elementary geometrical style, (2) to construct "the center and the radius" of the circle of Apollonius in the same way, (3) to present some properties of the circle of Apollonius. We described five individual methods on the construction of "the center and the radius" of the circle using elementary geometry, in the second part of this article. This is an important part of this article. Especially, the fourth method of this construction is equal to a means of finding out "both ends of the diameter" of the circle in the elementary geometrical style. The circle of Apollonius is one of the most excellent materials of teaching geometry in senior high school.