# Inversion in the Incircle What Is It About? A Mathematical Droodle

Explanation

### Inversion in the Incircle

The applet suggests the following theorem [Mathematical Gems II]:

The inverse images of the side lines of a triangle in its incircle are three circles of equal radii that concur at the incenter. The circle through their second points of intersection is none other than the inverse image of the circumcircle. It has the same radius.

R. Honsberger credits Arnold Emch (1916) with the following observation.

For a given $\Delta ABC,$ we are interested in the inversion in its incircle. The side lines (being tangent to the incircle) invert into the circles that pass through the incenter $I$ and still tangent to the side lines at the same points $A',$ $B',$ $C'$ of tangency, as the side lines. These circles have $IA',$ $IB',\;$ and $IC'$ as their diameters. The three are therefore equal.

Their second points of intersection correspond by the inversion to the vertices of $\Delta ABC.\;$ It follows that the circle through these three points is the inverse image of the circumcircle of $\Delta ABC.\;$ As we know, it has the same diameter as the other three circles, viz., $r$ - the inradius of $\Delta ABC.\;$ We have obtained a nice result: