Let a = (O, OB), b = (C, CB) and c = (D, DE) be three circles. (a) and (b) are tangent at B. All three have fixed radius and B moves on (a), remaining tangent to it all the time. For each position of B the bisectors of angle AOB are drawn and their intersections F, G with the radical axis of the circles (b) and (c) are defined. Line [FG] envelopes a conic (e).

JQ is the symmetric of radius OA with respect to the radical axis [FG] of circles (b) and (c). A point U, fixed on JQ describes, in general a curve, as B moves on circle (a). The picture shows the corresponding curve f(U), depending on U. For the various positions of U on JQ, this curve has the shape of an ovaloid, circle, limacon etc. Find its equation, using as parameter the position of the point U on JQ or its symmetric V, with respect to [FG].

In addition find the equation of the envelope of line JQ (the nephroid-like curve shown in the picture).

Produced with EucliDraw© |