TR-MO 1999 Problem 5

In an acute triangle $\vartriangle ABC$ with circumradius $R$ , altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$ , respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$ , respectively, to the circumcircle of triangle $\vartriangle DEF$ , prove that $\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$ .