The figure shows a triangle ABC with
the contact triangle DEF.
O, O_{1}, O_{2}, O_{3}, and O_{4} are the
incenters of triangles ABC, ADF, BDE, CEF, and DEF, and r, r_{1},
r_{2}, r_{3}, and r_{4} are the inradii,
respectively. Prove (1) O_{1}, O_{2}, O_{3} are the midpoints of the arcs DF, DE, and EF
respectively, and
(2) r_{1} + r_{2} + r_{3} + r_{4}
= 2.r
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