Inversion with Negative Power: What is this about?
A Mathematical Droodle
What if applet does not run? |
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Copyright © 1996-2018 Alexander BogomolnyInversion with Negative Power
What if applet does not run? |
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:
All circles through two diametrically opposite points on a given circle t and point A not at the center of t also pass through another point A'. |
(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.
ΔMCN is right, therefore
(1) | MO×NO = CO^{2} = OR^{2}. |
Let A' be the second point of intersection of OA with w. By the Intersecting Chords Theorem,
(2) | MO×NO = OA×OA'. |
Combining (1) and (2) gives
(3) | OA×OA' = OR^{2}, |
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
(4) | OA×OA' = -OR^{2}. |
The latter defines inversion with a negative power. The applet demonstrates the geometric meaning of such a transformation. Inversion in t with the power
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Copyright © 1996-2018 Alexander Bogomolny