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2 triangles

Source: Moldova TST 2005

April 10, 2005
geometrycircumcircleinradiusinequalitiestrigonometrytrig identitiesLaw of Sines

Problem Statement

Let ABCABC and A1B1C1A_{1}B_{1}C_{1} be two triangles. Prove that aa1+bb1+cc1≤3R2r1\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}, where a=BCa = BC, b=CAb = CA, c=ABc = AB are the sidelengths of triangle ABCABC, where a1=B1C1a_{1}=B_{1}C_{1}, b1=C1A1b_{1}=C_{1}A_{1}, c1=A1B1c_{1}=A_{1}B_{1} are the sidelengths of triangle A1B1C1A_{1}B_{1}C_{1}, where RR is the circumradius of triangle ABCABC and r1r_{1} is the inradius of triangle A1B1C1A_{1}B_{1}C_{1}.
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