Inversion with Negative Power: What is this about?
A Mathematical Droodle

Explanation

The applet is supposed to remind of the symmetry in circle - inversion. The inverse image A' of a point A has the property that all circles perpendicular to the circle of inversion t and passing through A invariably pass through A'.

The current situation is slightly different:

(In the applet the circle t is defined by its draggable center O and point R.)

Proof

Let CD be a diameter of t, w the circle through C, D and A, and let MN is the diameter of w that passes through O. CD is then perpendicular to MN.

MCN is right, therefore

Let A' be the second point of intersection of OA with w. By the Intersecting Chords Theorem,

Combining (1) and (2) gives

which appears exactly like the inversion identity, but is not quite the same. The difference is in that formerly the points A and A' lay on the same side from O, while now O separates the two. If we consider OA and OA' as signed segments, then (3) should be corrected to

The latter defines inversion with a negative power. The applet demonstrates the geometric meaning of such a transformation. Inversion in t with the power p < 0 is equivalent to the inversion with the power |p| and subsequent reflection in O.

Inversion - Introduction

• Angle Preservation Property
• Apollonian Circles Theorem
• Archimedes' Twin Circles and a Brother
• Bisectal Circle
• Chain of Inscribed Circles
• Circle Inscribed in a Circular Segment
• Circle Inversion: Reflection in a Circle
• Circle Inversion Tool
• Feuerbach's Theorem: a Proof
• Four Touching Circles
• Hart's Inversor
• Inversion in the Incircle
• Inversion with a Negative Power
• Miquel's Theorem for Circles
• Peaucellier Linkage
• Polar Circle
• Poles and Polars
• Radical Axis of Circles Inscribed in a Circular Segment
• Steiner's porism
• Stereographic Projection and Inversion
• Tangent Circles and an Isosceles Triangle
• Tangent Circles and an Isosceles Triangle II
• Three Tangents, Three Secants