A line l intersects the segment AB perpendicular to C. Three circles are drawn successively with AB,AC and BC as the diameter. The largest circle intersects l in D. The segments DA and DB still intersect the two smaller circles in E and F.
a. Prove that quadrilateral CFDE is a rectangle.
b. Prove that the line through E and F touches the circles with diameters AC and BC in E and F.[asy]
unitsize (2.5 cm);pair A, B, C, D, E, F, O;O = (0,0);
A = (-1,0);
B = (1,0);
C = (-0.3,0);
D = intersectionpoint(C--(C + (0,1)), Circle(O,1));
E = (C + reflect(A,D)*(C))/2;
F = (C + reflect(B,D)*(C))/2;draw(Circle(O,1));
draw(Circle((A + C)/2, abs(A - C)/2));
draw(Circle((B + C)/2, abs(B - C)/2));
draw(A--B);
draw(interp(C,D,-0.4)--D);
draw(A--D--B);dot("A", A, W);
dot("B", B, dir(0));
dot("C", C, SE);
dot("D", D, NW);
dot("E", E, SE);
dot("F", F, SW);
[/asy] geometryrectangletangent circles