MathDB

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

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

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]

Problem Statement