Circles (A47a) and (A47b): The eyeball pair


Consider the tangent from O1 to (O2). It meets (O1) in a point X1. Similarly the tangent from O2 to (O1) meets (O2) in a point X2. According to the eyeball theorem X1 and X2 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 A47a and A47b are the feet of perpendiculars from X1 and X2 to AB, then the circles (A47a) and (A47b) through X1 and X2 respectively are Archimedean. [Phil Todd, 18 Sep 2007]

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