From P, take a random line that cut the smaller ciricle, say the chord's length is a. Make a chord with length a at the bigger circle. Say the center of the bigger circle is O Make another circle X, concentric with the bigger circle, with radius twice the length of the distance of the midpoint of the chord to O. Let Q at X so PO= PQ Take a line through P and the midpoint of the OQ

Consider the circles (K,r) and (L,R) with r<R. Consider also the line e which passes through P. Assume $ x = KK^' \bot e\;and\;y = LL^' \bot e.$ Basically we want: $ y^2 - R^2 = x^2 - r^2 \Rightarrow y^2 - x^2 = R^2 - r^2 = c^2 \;or\;\left( {y - x} \right)\left( {y + x} \right) = c^2$ (c constant). If $ KK_1^' \bot LL^' \;and\;K^{''}$ the symmetric point of $ K_1^'$ with axis symmetry the line e, which moves (the point $ K^{''}$) along the constant circle (1). If $ K^{''} \bot LK$ then $ LS \cdot LK = c^2$, that means S is constant. Therefore $ SK^{''}$ determines, finally, a constant line (2). The common point of circle (1) with the line (2) are giving the point $ K^{''} ....$ S.E.Louridas