1989 AIME Question 15

Point $P$ is inside triangle $ABC.$ Line segments $APD,$ $BPE,$ and $CPF$ are drawn with $D$ on $BC,$ $E$ on $AC,$ and $F$ on $AB$ (see the figure). Given that $AP=6,$ $BP=9,$ $PD=6,$ $PE=3,$ and $CF=20,$ find the area of triangle $ABC.$