A circle (of radius a), a line, and a point are mapped by inversion into two concentric circles and the center of those concentric circles. If the distance from the given circle's center to the line is b, then what is the distance from the point to the line?

Inversion Defined:

Let O be the center of a circle of radius k. An inversion with respect to circle O is a mapping f:R² -> R² such that for all P in R² (not O), P' = f(P) lies on ray OP and
|OP'||OP| = k².

See www.geocities.com/bractals/inv.jpg

for graphical description of inversion.

(In reply to problem by Charlie)

For the reason given in my earlier post, I do not think Eric's "result" is correct. 

I agree with Charlie and Eric that we can show using the point and line only that the distance from the centre of the circle of inversion O to the line L |OL| = |PL| the distance from the given point P to the line.

We have got this far using only the point and line.  Adding in the circle C, I  think gives us enough to solve using a and b, (both of which refer to the given circle) obviating the need for k.

Posted by goFish on 2006-02-22 11:05:26