As shown in the figure, triangle ABC is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle ABC.[asy]
size(200);
pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C);
draw(F--C--A--B--C^^A--D^^B--E);
label("A", A, SW);
label("B", B, SE);
label("C", C, N);
label("84", centroid(H, C, E), fontsize(9.5));
label("35", centroid(H, B, D), fontsize(9.5));
label("30", centroid(H, F, B), fontsize(9.5));
label("40", centroid(H, A, F), fontsize(9.5));[/asy] geometryratioAsymptotealgebrasystem of equations