Circular Motion without Physics
Let \(ABC\) be a triangle. Let \(I\) be its incenter. Let \(L, M, N\) be the circumcenters of triangles \(BIC, AIC, AIB\), respectively. What is the sum of the powers of \(L, M, N\) with respect to the circumcircle of \(\triangle ABC\)?
Note:The power of point P to circle \(\omega\) with radius \(r\) and center \(O\) is \(OP^2 - r^2\).