Circles Γ

_{1}and Γ

_{2}( with centers O

_{1}and O

_{2}respectively ) intersect

at points A and B such that neither center lies on or inside the other circle.

Lines m

_{A}and m

_{B}are tangent to circle Γ

_{1}at points A and B respectively.

Distinct points C and D lie on Γ

_{2}∩ H(A,m

_{B}) ∩ H(B,m

_{A}), where H(X,m

_{Y})

denotes the open half-plane determined by the line m

_{Y}and does not contain point X.

Rays CA, CB, DA, and DB intersect Γ

_{1}at points C

_{A}, C

_{B}, D

_{A}, and D

_{B}

respectively.

I = C

_{A}D

_{A}∩ C

_{B}D

_{B}

and

J = C

_{A}C

_{B}∩ D

_{A}D

_{B}

Prove that points I, J, and O

_{1}are collinear.