# Circles (A_{50a}) and (A_{50a}): The 50^{th} 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 O_{1} to (O_{2}) and O_{2} to (O_{1}) intersecting at H.
Create J in such a way that triangle JO_{1}O_{2} has H as its incenter.
JO_{1} meets (O_{1}) in K and the circle with diameter CK is Archimedean circle (A_{50a}).
JO_{2} meets (O_{1}) in L and the circle with diameter CL is Archimedean circle (A_{50b}).
[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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