AMC10 2006 10B, Problem 24

Circles with centers $O$ and $P$ have radii $2$ and $4$, respectively, and are externally tangent. Points $A$ and $B$ on the circle with center $O$ and points $C$ and $D$ on the circle with center $P$ are such that $AD$ and $BC$ are common external tangents to the circles. What is the area of the concave hexagon $AOBCPD$? [asy]size(200);defaultpen(linewidth(0.8)); pair X=(-6,0), O=origin, P=(6,0), B=tangent(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2); pair top=X+15*dir(X--A), bottom=X+15*dir(X--B); draw(Circle(O, 2)^^Circle(P, 4)); draw(bottom--X--top); draw(A--O--B^^O--P^^D--P--C); pair point=X; label("$2$", midpoint(O--A), dir(point--midpoint(O--A))); label("$4$", midpoint(P--D), dir(point--midpoint(P--D))); label("$O$", O, SE); label("$P$", P, dir(point--P)); pair point=O; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); pair point=P; label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); fill((-3,7)--(-3,-7)--(-7,-7)--(-7,7)--cycle, white);[/asy]

Solution: When we see this problem, it practically screams similar triangles at us. Extend $OP$ to the left until it intersects lines $AD$ and $BC$ at point $E$. Triangles $EBO$ and $ECP$ are similar, and by symmetry, so are triangles $EAO$ and $EDP$. Then, the area of kite $EAOB$ is just $4 \sqrt{2} \times 2$ and the area of kite $EDPC$ is $8 \sqrt{2} \times 4$ (using our similarity ratios). The difference of these yields the area of hexagon $AOBCPD$ or $24\sqrt{2} \Longrightarrow \fbox{24\sqrt{2}}$.

Answer: 24\sqrt{2}