CutUnderGivenAngle

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.

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.

Cut under given angle

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