[alogo] Inverting on family members

Consider a circle bundle (I) of circles f.e. generated by two given circles c1 and c2. Consider also an arbitrary circle c. Find the envelope of circles (c') resulting by inverting (c) with respect to member circles d of the family (II) orthogonal to (I).

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]

The solution is found by determining the two circle-members c3, c4 of family (I) that are tangent to c (see TangentMember.html ). Since inversion with respect to (d) leaves all circles of family (I) invariant c' will contact both c3 and c4.
Notice that the centers C' of circles c' describe an ellipse (f) with foci at C3, C4 (centers of c3, c4) and major axis 2*a = r3+r4, r3, r4 being the raddii of circle c3, c4 correspondingly (supply trivial proof).

See Also


Return to Gallery

Produced with EucliDraw©