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We investigate two interesting special cases of the classical Apollo-nius problem, and then apply these to the tritangent circles of a triangle to find pair of perspective (or homothetic) triangles. Some new triangle centers are constructed. 1. An interesting construction We begin with a simple construction of a special case of the classical Apollonius problem. Given two circles O(r), O ′ (r ′) and an external tangent L, to construct a circle O 1 (r 1) tangent to the circles and the line, withfatcat:mlyynj6iivgklmuayfa4niaspa