May 15th, 2017, 06:16 PM  #1 
Newbie Joined: May 2017 From: Buffalo NY Posts: 2 Thanks: 0  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.] 
May 20th, 2017, 04:08 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,159 Thanks: 866 
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: 88 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: 

Tags 
circles, euclidean geometry, orthogonal, proving 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How do you show a MÃ¶bius transformation maps circles to circles and lines?  math93  Geometry  0  November 3rd, 2015 02:43 PM 
RiDo Circles. Sin & Cos Circles  RiDo  Algebra  2  June 21st, 2012 01:31 AM 
Proving integral with integrands f(x) and g(x) is orthogonal  1bh  Calculus  2  June 16th, 2009 07:17 AM 