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, O1, O2, and O3 are the circumcenters of
triangles ABC, AHM, BDE, and CFG respectively. I1, I2, and I3
are the incenters of triangles AHM, BDE, and CFG respectively. If
d = OI, d1 = O1I1, d2=O2I2, and O3 = O3I3, (1) prove that d, d1,
d2, and d3 are parallel, (2) prove that d = d1 + d2 + d3.
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