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Mediterranean M.C. 2010 Problem 3

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July 12, 2010
geometrycircumcircleinradiusgeometric transformationreflectionincenterEuler

Problem Statement

Let A′∈(BC),A'\in(BC), B′∈(CA),C′∈(AB)B'\in(CA),C'\in(AB) be the points of tangency of the excribed circles of triangle △ABC\triangle ABC with the sides of △ABC.\triangle ABC. Let R′R' be the circumradius of triangle △A′B′C′.\triangle A'B'C'. Show that R′=12r2R(2R−ha)(2R−hb)(2R−hc) R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)} where as usual, RR is the circumradius of △ABC,\triangle ABC, r is the inradius of △ABC,\triangle ABC, and ha,hb,hch_{a},h_{b},h_{c} are the lengths of altitudes of △ABC.\triangle ABC.
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