My Math Forum Proving orthogonal circles

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May 15th, 2017, 06:16 PM   #1
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Proving orthogonal circles

Starting from the circle and the point A as in the previous part, we construct Q as before, and then construct the line l through Q perpendicular to OA. Show that for any point P on l, the circle with center P and radius PA is orthogonal to the original circle. Please see the attached image to view figure
[Previous problem says the following (may or may not be helpful): Given a circle with center O and a point A is not the same as O inside the circle, we construct the line perpendicular to OA at A and denote by X one of the intersections of that line with the circle. The tangent line to the circle at X then intersects OA at point C. We let Q be the midpoint of AC.]
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 May 20th, 2017, 04:08 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 Look at this: https://en.wikipedia.org/wiki/Inversive_geometry (It doesn't help to post the same thing three times!)
 May 22nd, 2017, 08:23 AM #3 Member   Joined: Jan 2016 From: Athens, OH Posts: 89 Thanks: 47 As Country Boy suggests, this is easily solved with inversion with respect to a circle. Here's the diagram and solution that relies on a simple theorem:

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