Circles (A50a) and (A50a): The 50th pair

Definition

The circle through C tangent to the tangent from A to the circle with center B through C is Archimedean. Similarly the circle through C tangent to the tangent from B to the circle with center A through C is Archimedean. [FvL, 27 sep 2007].

Properties

Consider the tangents from O1 to (O2) and O2 to (O1) intersecting at H. Create J in such a way that triangle JO1O2 has H as its incenter. JO1 meets (O1) in K and the circle with diameter CK is Archimedean circle (A50a). JO2 meets (O1) in L and the circle with diameter CL is Archimedean circle (A50b). [Miguel Ochoa Sanchez (conjecture) and Emmanuel Antonio José García (proof), june 2014]
Source: A. Bogomolny, Shedding Light on the Ball for Eyeballing from Interactive Mathematics Miscellany and Puzzles http://www.cut-the-knot.org/pythagoras/LightingTheBall.shtml

See also Archimedean circle 62: The Sanchez-García triplet circle.

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