Another Hexagon Problem?
Source: 2022 AIME II Problem 15
February 17, 2022
AMCAIMEAIME II
Problem Statement
Two externally tangent circles and have centers and , respectively. A third circle passing through and intersects at and and at and , as shown. Suppose that , , , and is a convex hexagon. Find the area of this hexagon.
[asy]
import geometry;
size(10cm);
point O1=(0,0),O2=(15,0),B=9*dir(30);
circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B);
point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0];
filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black);
draw(w1);
draw(w2);
draw(O1--O2,dashed);
draw(o);
dot(O1);
dot(O2);
dot(A);
dot(D);
dot(C);
dot(B);
label("",8*dir(110),SW);
label("",5*dir(70)+(15,0),SE);
label("",O1,W);
label("",O2,E);
label("",B,N+1/2*E);
label("",A,N+1/2*W);
label("",C,S+1/4*W);
label("",D,S+1/4*E);
label("",midpoint(O1--O2),N);
label("",midpoint(C--D),N);
label("",midpoint(A--B),S);
label("",o.C+(o.r-1)*dir(270));
[/asy]