# Circles (A_{47a}) and (A_{47b}): The eyeball pair

### Definition

Consider the tangent from O_{1} to (O_{2}). It meets (O_{1}) in a point X_{1}.
Similarly the tangent from O_{2} to (O_{1}) meets (O_{2}) in a point X_{2}.
According to the eyeball theorem X_{1} and X_{2} are at equal distance from AB, more precisely
this distance is equal to the radius of an Archimedean circle
(see Cut the knot, The Eyeball Theorem, proof 2).
Hence if A_{47a} and A_{47b} are the feet of perpendiculars from X_{1} and X_{2}
to AB, then the circles (A_{47a}) and (A_{47b}) through X_{1} and X_{2}
respectively are Archimedean. [Phil Todd, 18 Sep 2007]

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