Van Lamoen circle

In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle ${\displaystyle T}$. It contains the circumcenters of the six triangles that are defined inside ${\displaystyle T}$ by its three medians.[1][2]

The van Lamoen circle through six circumcenters ${\displaystyle A_{b}}$, ${\displaystyle A_{c}}$, ${\displaystyle B_{c}}$, ${\displaystyle B_{a}}$, ${\displaystyle C_{a}}$, ${\displaystyle C_{b}}$

Specifically, let ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$ be the vertices of ${\displaystyle T}$, and let ${\displaystyle G}$ be its centroid (the intersection of its three medians). Let ${\displaystyle M_{a}}$, ${\displaystyle M_{b}}$, and ${\displaystyle M_{c}}$ be the midpoints of the sidelines ${\displaystyle BC}$, ${\displaystyle CA}$, and ${\displaystyle AB}$, respectively. It turns out that the circumcenters of the six triangles ${\displaystyle AGM_{c}}$, ${\displaystyle BGM_{c}}$, ${\displaystyle BGM_{a}}$, ${\displaystyle CGM_{a}}$, ${\displaystyle CGM_{b}}$, and ${\displaystyle AGM_{b}}$ lie on a common circle, which is the van Lamoen circle of ${\displaystyle T}$.[2]