## A Circle Related to Incenter and CircumcenterWhat Is This About? A Mathematical Droodle

5 November 2015, Created with GeoGebra

Explanation

### 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.

5 November 2015, Created with GeoGebra

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 CI' = (a + b - c)/2. CO' = b/2. Thus O'I' = (a - c)/2. But, since ON is a chord in I(O) with center I, OL = LN. And since O'I' = OL, ON = a - c = CD. Similarly we can show that OM = CE. The triangles CDE and ONM are congruent by SAS.