Problem

Source: Iranian National Olympiad (3rd Round) 2003

Tags: geometry, incenter, circumcircle, trapezoid, geometric transformation, reflection, geometry proposed



Suppose that $ M$ is an arbitrary point on side $ BC$ of triangle $ ABC$. $ B_1,C_1$ are points on $ AB,AC$ such that $ MB = MB_1$ and $ MC = MC_1$. Suppose that $ H,I$ are orthocenter of triangle $ ABC$ and incenter of triangle $ MB_1C_1$. Prove that $ A,B_1,H,I,C_1$ lie on a circle.