Draft:Mansion's theorem

• Comment: We need more than one source. Qcne (talk) 16:00, 23 June 2024 (UTC)

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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 $BM = IM$.

External Links

• [Mansion Theorem at Cut-the-Knot](https://www.cut-the-knot.org/m/Geometry/MansionTheorem.shtml)