All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
Circumcenters and Vectors (Posted on 2017-12-31) Difficulty: 3 of 5

  
1) Let I be the incenter of acute ΔABC and A' the circumcenter of ΔIBC.

     Prove that A, I, and A' are collinear.

2) Let B' and C' be the circumcenters of triangles ICA and IAB respectively.

     Prove that the sum of the vectors IA', IB', and IC' is the zero vector 0
     if and only if ΔABC is equilateral.
  

No Solution Yet Submitted by Bractals    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Analytic solution Comment 2 of 2 |
(In reply to Analytic solution by Jer)

You lost me at "thus I=(0,0), A=(a,0), B=(b,0)".


Line AB is tangent to the incircle, but "thus ..." implies that
it passes through the center of the incircle.

BTW did you notice that the triangles ABC and A'B'C' have the
same circumcircle and |IX| * |IX'| = 2*r*R
where r and R are the inradius and circumradius of triangle ABC.

Edited on January 2, 2018, 5:38 pm
  Posted by Bractals on 2018-01-02 13:51:38

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (5)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2018 by Animus Pactum Consulting. All rights reserved. Privacy Information