Draft:Mansion's theorem

In Euclidean geometry, the Mansion theorem is a theorem concerning the circumcircle, incenter, and excenters of a triangle.

The statement of the theorem

Let $I$ be the incenter, $I_a$ an excenter of $\triangle ABC$ aposite to $A$. The intersecting point of $II_a$ and the circumcircle $M$ is the midpoint of the segment of $II_a$. Moreover, the points $I, I_a, B, C$ is on the circle with diameter $II_a$.

Proof

Consider a triangle $ABC$. Let $\Omega$ be the circumcircle of it. Let $I$ and $I_a$ be the incenter and the excenter aposite to $A$. Then by the property of the inscribed angle, $\angle IBM = \angle IBC+\angle CBM = \angle IBA + \angle CAI = \angle IBA + \angle IAB = \angle BIM$. Therefore $BM = IM$.