Properties of Circle Through the Incenter
Circle through two vertices and the incenter of a triangle has many attractive properties.
Below I list a few (that I know of) with proofs or links to the pages that establish these properties. Every one's welcome to suggest either properties not listed here or alternative proofs to the one that are.
The center of $(ABI)$ lies on the bisector of $\angle ACB.$
Let $X$ be the second intersection of $BC$ with $(ABI),$ $Y$ the second intersection of $AC$ with $(ABI).$
Point $E$ on circle $(ABI)$ through $A,B,I$ has the property that $BE=AB.$
Then $\angle ABE=\angle ACB.$
The center $O_c$ of $(ABI)$ lies on the circumcircle $(ABC).$
Let $BE$ be a chord in $(ABI)$ such that $BE=AB.$ Let $O$ be the circumcenter of $\Delta ABC.$
Then $BE\perp BO.$