16. En $\triangle ABC$ , las medianas $AD$ y $CE$ tienen longitudes $18$ y $27$ , respectivamente, y $AB = 24$ . Prolongue $CE$ hasta intersecar la circunferencia circunscrita de $\triangle ABC$ en $F$ . Halle el área de $\triangle AFB$ . [asy] import graph; size(200); // Coordinates based on median intersection (Centroid G at origin) pair G = (0,0); pair A = (0,12); // AG = 12 (2/3 of median 18) pair D = (0,-6); // GD = 6 (1/3 of median 18) pair C = (18,0); // GC = 18 (2/3 of median 27) pair E = (-9,0); // GE = 9 (1/3 of median 27) // E is the midpoint of AB, so B = 2E - A pair B = (-18,-12); // Calculate circumcenter (O) of ABC // Intersection of perpendicular bisectors of AB and AC pair midAB = (A+B)/2; pair midAC = (A+C)/2; pair perpAB = rotate(90, midAB) * A; pair perpAC = rotate(90, midAC) * A; pair O = extension(midAB, perpAB, midAC, perpAC); real R = abs(A-O); // Draw circumcircle draw(circle(O, R)); // Line CE intersects circle at F. // F lies on the line passing through C(18,0) and E(-9,0), which is the x-axis. // Intersection of x-axis (y=0) and (x-Ox)^2 + (y-Oy)^2 = R^2 real Fx = O.x - sqrt(R^2 - O.y^2); // The point on the left side pair F = (Fx, 0); // Draw lines draw(A--B--C--cycle); draw(A--D); draw(C--F); draw(A--F, dashed + blue); draw(B--F, dashed + blue); // Labels dot("$A$", A, N); dot("$B$", B, SW); dot("$C$", C, SE); dot("$D$", D, S); dot("$E$", E, NW); dot("$F$", F, W); dot("$G$", G, NE); // Adding length labels to match problem info label("$12$", (A+E)/2, W); label("$12$", (B+E)/2, W); label("$18$", (G+C)/2, S); label("$9$", (G+E)/2, S); [/asy] Pi-Infinity