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June 2nd, 2019, 10:41 AM  #1 
Senior Member Joined: Jan 2012 Posts: 137 Thanks: 2  Circumradius of an especial triangle
Hello, There is a triangle ABC, the orthocenter of which is the point P. How can we find the circumradius of the triangle PBC in terms of R (where R is the circumradius of the triangle ABC)? Thanks. Last edited by skipjack; June 2nd, 2019 at 11:35 AM. 
June 2nd, 2019, 12:15 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
They're equal.

June 2nd, 2019, 12:34 PM  #3  
Math Team Joined: Jul 2011 From: Texas Posts: 2,940 Thanks: 1545  Quote:
Seems like the circumradii are congruent. Maybe you can see a way to prove it?  
June 2nd, 2019, 02:44 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
BC is a chord of both circumcircles. Consider the angles it subtends at A and P.

June 3rd, 2019, 01:55 AM  #5 
Senior Member Joined: Jan 2012 Posts: 137 Thanks: 2 
I guess we can also prove using Sine rule...but I am not getting exactly how to do...Can u elaborate that? Thx.

June 3rd, 2019, 04:12 AM  #6 
Senior Member Joined: Jan 2012 Posts: 137 Thanks: 2  
June 3rd, 2019, 10:41 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
The angles I mentioned are supplementary, so they have equal sine, which makes proof by the sine rule easy.

June 3rd, 2019, 11:50 AM  #8 
Senior Member Joined: Jan 2012 Posts: 137 Thanks: 2  

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