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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    
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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

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