My Math Forum  

Go Back   My Math Forum > High School Math Forum > Trigonometry

Trigonometry Trigonometry Math Forum


Thanks Tree2Thanks
  • 1 Post By skeeter
  • 1 Post By skipjack
Reply
 
LinkBack Thread Tools Display Modes
June 2nd, 2019, 10:41 AM   #1
Senior Member
 
happy21's Avatar
 
Joined: Jan 2012

Posts: 140
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.
happy21 is offline  
 
June 2nd, 2019, 12:15 PM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 20,969
Thanks: 2219

They're equal.
skipjack is offline  
June 2nd, 2019, 12:34 PM   #3
Math Team
 
skeeter's Avatar
 
Joined: Jul 2011
From: Texas

Posts: 3,016
Thanks: 1600

Quote:
Originally Posted by happy21 View Post
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)?
Two constructions attached. D is the circumcenter of ABC.

Seems like the circumradii are congruent. Maybe you can see a way to prove it?
Attached Images
File Type: jpg ortho1.jpg (45.2 KB, 4 views)
File Type: jpg ortho2.jpg (43.3 KB, 3 views)
Thanks from happy21
skeeter is offline  
June 2nd, 2019, 02:44 PM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 20,969
Thanks: 2219

BC is a chord of both circumcircles. Consider the angles it subtends at A and P.
skipjack is offline  
June 3rd, 2019, 01:55 AM   #5
Senior Member
 
happy21's Avatar
 
Joined: Jan 2012

Posts: 140
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.
happy21 is offline  
June 3rd, 2019, 04:12 AM   #6
Senior Member
 
happy21's Avatar
 
Joined: Jan 2012

Posts: 140
Thanks: 2

Quote:
Originally Posted by skeeter View Post
Two constructions attached. D is the circumcenter of ABC.

Seems like the circumradii are congruent. Maybe you can see a way to prove it?
Can u prove it in steps also? Thx.
happy21 is offline  
June 3rd, 2019, 10:41 AM   #7
Global Moderator
 
Joined: Dec 2006

Posts: 20,969
Thanks: 2219

The angles I mentioned are supplementary, so they have equal sine, which makes proof by the sine rule easy.
Thanks from happy21
skipjack is offline  
June 3rd, 2019, 11:50 AM   #8
Senior Member
 
happy21's Avatar
 
Joined: Jan 2012

Posts: 140
Thanks: 2

Quote:
Originally Posted by skipjack View Post
The angles I mentioned are supplementary, so they have equal sine, which makes proof by the sine rule easy.
Ohk...I try that way.
happy21 is offline  
Reply

  My Math Forum > High School Math Forum > Trigonometry

Tags
circumradius, especial, triangle



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
is orthic triangle same as pedal triangle? randomgamernerd Geometry 1 November 16th, 2015 08:09 PM
Triangle circumradius problem USAMO Reaper Geometry 4 January 29th, 2015 10:33 PM
Triangle w0w Algebra 3 June 24th, 2013 11:59 AM
Triangle help. Please captonion Algebra 6 June 24th, 2010 08:46 AM
Moller Triangle to Triangle Collisions 1101 Algebra 0 June 19th, 2009 10:56 PM





Copyright © 2019 My Math Forum. All rights reserved.