## A Circle Related to Incenter and Circumcenter

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Copyright © 1996-2018 Alexander Bogomolny

### A Circle Related to Incenter and Circumcenter

A problem from the Russian *Kvant* reveals additional properties of a construction that has seemingly been scrutinized at this site (1, 2, 3, 4):

As usual, let I and O be the incenter and circumcenter, respectively, of triangle ABC. Suppose the side AB is laid off along each of the other two sides to give points D and E so that

EA = AB = BD. |

Let I(O) be the circle through O with center at I. If chords OM||BC and ON||AC then ΔCDE = ΔONM.

As usual, a, b, c denote the lengths of sides BC, AC, and AB.

First of all, angles DCE and NOM have parallel sides and are therefore equal. Let I' and O' be the feet of perpendiculars from I and O onto AC and L the intersection of II' and ON. O' is the midpoint of AC; I' is the point of tangency with AC of the incircle of ΔABC. We know that

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny