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5

Part of 1999 Turkey MO (2nd round)

Problems(1)

Turkey NMO 1999, P-5, nice geometric inequality on altitudes

Source:

12/23/2010
In an acute triangle △ABC\vartriangle ABC with circumradius RR, altitudes AD‾,BE‾,CF‾\overline{AD},\overline{BE},\overline{CF} have lengths h1,h2,h3{{h}_{1}},{{h}_{2}},{{h}_{3}}, respectively. If t1,t2,t3{{t}_{1}},{{t}_{2}},{{t}_{3}} are lengths of the tangents from A,B,CA,B,C, respectively, to the circumcircle of triangle △DEF\vartriangle DEF, prove that ∑i=13(tihi)2≤32R\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R.
inequalitiesgeometrycircumcircletrigonometrygeometry proposed