Abstract
Given a ratio α ∶ β ∶ γ, α>γ>0, and a triangle ΔABC, on the sides\(\overline {BC} ,\overline {CA} \) and\(\overline {AB} \), using ratiosβ ∶ γ, α ∶ γ and α ∶ β, three circles of Apollonius are denned. In this paper, we will show that the three centers are collinear, the circles are coaxal and develop a necessary and sufficient condition that these circles intersect. J. A. Hoskins, W. D. Hoskins and R. G. Stanton obtained these results in a recent paper using algebraic computation. Our aim is to establish all these results using only results from elementary Euclidean geometry and thereby uncovering more geometric insights and avoid lengthy calculations.
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