In the figure below, given a triangle
ABC and the incircle of center I (inscribed circle), DE, FG, and HM are
tangent to the incircle I and parallel to AC, AB, and BC
respectively. O, O_{1}, O_{2}, and O_{3} are the circumcenters of
triangles ABC, AHM, BDE, and CFG respectively. I_{1}, I_{2}, and I_{3}
are the incenters of triangles AHM, BDE, and CFG respectively. If
d = OI, d_{1} = O_{1}I_{1}, d_{2}=O_{2}I_{2}, and O_{3} = O_{3}I_{3}, (1) prove that d, d_{1},
d_{2}, and d_{3} are parallel, (2) prove that d = d_{1} + d_{2} + d_{3}.
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