Point P is in the interior of \triangle A B C. The side lengths of A B C are A B=7, B C=8, C A=9. The three foots of perpendiculars from P to sides B C, C A, A B are D, E, F respectively. Suppose the minimal value of \frac{B C}{P D}+\frac{C A}{P E}+\frac{A B}{P F} can be written as \frac{a}{b} \sqrt{c}, where \operatorname{gcd}(a, b)=1 and c is square free, calculate a b c.
