[alogo] Cut under given angle

Given two points A,B and a circle (c). To construct a circle (c') passing through {A,B} and cutting (c) under a given angle.

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Consider all the circles (c') passing through A and intersecting the given circle (c) under the given angle. Consider also the inversion F on the circle (d) centered at A and orthogonal to (c). By this inversion the circle (c') transforms to a line F(c'). The condition of the intersection of {c',c} under a fixed angle, translates to the tangency of lines F(c') to a circle c0, concentric to c. Point B transforms by F to B' and the problem reduces to the one of drawing tangents from B' to circle c0.
The discussion of the various possibilities and conditions of existence of solutions is fairly straighforward.
More interesting seems to me the locus of the centers of circles (c'), which is a hyperbola (h). Hyperbola (h) is the homothetic of another hyperbola (h') resulting by inverting a certain limacon (m) with respect to A. The limacon is the pedal of circle c0 with respect to A, i.e. the locus of the projections of A on the tangents to c0.
By well known properties of the limacon (see Limacon.html ), the inverse of this curve with respect to circles centered at A are hyperbolas. Thus, the inverses F of E describe an hyperbola h' and G, being the middle of AF, describes correspondingly a homothetic to h' hyperbola h.
Reversing the argument we obtain a characterization of the hyperbola as the locus of centers of circles passing through a fixed point A and intersecting a fixed circle (c) under a fixed given angle. Point A is a focus of this hyperbola.

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