Here we show that circles not passing through the center of the inverting circle invert to other circles not passing through the center of the inverting circle. |
First we establish a property of circles: |
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Now suppose C is a circle outside the inverting circle. |
A point P on C inverts to a point P' with OP*OP' = r2, where r is the radius of the inverting circle. Consequently, |
OP' = r2/OP |
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Let Q be the other point of C on the line through O and P. |
Then by the result above, OQ*OP = k, so |
OQ = k/OP |
Combining the two red equations gives |
OP'/OQ = r2/k |
That is, |
OP' = (r2/k)OQ |
As Q traces out the points of C, P' also traces out a circle, because OP' is
OQ, scaled by a constant factor |
An algebraic derivation of the formula for the center and radius of the inverted circle is found here. |
Return to properties of inversion.