Circumcenter on Angle Bisector: What Is It About?
A Mathematical Droodle

Explanation

Copyright © 1996-2009 Alexander Bogomolny

The applet below provides an illustration to a problem from an outstanding collection by T. Andreescu and R. Gelca:

In ΔABC let I be the incenter. Prove that the circumcenter of ΔBIC lies on AI.

I is the meeting point of the angle bisectors of ΔABC. In particular, AI is the angle bisector of A. The excenter E opposite A, is the point of concurrency of AI and the exterior bisectors of angles at B and C. The latter are perpendicular to the angle bisectors BI and CI so that the quadrilateral BICE is cyclic and IE is a diameter of its circumcircle. Naturally, this is also the circumcircle of ΔBIC. Its center is the midpoint of IE and since A, I, and E are collinear, the conclusion follows.

It is also clear that the circumcenter O of ΔBIC lies on the circumcircle of ΔABC.

References

1. T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, 5^th printing, 1.2.9 (p. 9)

Copyright © 1996-2009 Alexander Bogomolny