David Little
Mathematics Department
Penn State University
Eberly College of Science
University Park, PA 16802
Office: 403 McAllister
Phone: (814) 865-3329
Fax: (814) 865-3735

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Inversion of a Point across a Circle
Our goal will be to construct the inversion of the blue point, P, with respect to the circle centered at the green point.

To do so, first pick any point, Q, inside the circle. Next, draw a circle centered at P that goes through the point Q. This circle must intersect the original circle in exactly two points. Mark these two points of intersection.

Next, draw two lines. Each line should go through P and one of the points of intersection you just constructed. These two lines should each intersect the original circle in two places. Mark the points of intersection of these two lines and the original circle. These four points of intersection form the vertices of a trapezoid.

And finally, draw two line segments that join opposite vertices of the trapezoid. These two line segments intersect at a point P', which is called the inversion of P. The inversion point P' has the following property. Let D represent the distance from P to the green point, let d represent the distance from P' to the green point and let r represent the radius of the circle centered at the green point. Then

Dd = r2.
Can you explain why?

Once you have performed the above construction... consider a circle and a point on a line. move the point on the line, what curve is traced out by it's inversion do the same but with another circle