The common tangents AB and CD to two circles with centre O and O' intersect at E between their centres.Prove that the points O,E,and O'are collinear. Plz explain this question in most easy way??fast??

∠AEC = ∠DEB (Vertically Opposite Angle)
join OA and OC
so in triangle OAE and triangle OCE we have,
OA=OC (radii of same circle)
OE=OE (common)
∠OAE=∠OCE (90° as the tangent is always perpendicular to the radius at the                            point of contact)
∴ ΔOAE ≡ ΔOCE
so ∠AEO = ∠CEO (CPCT)
similarly for other circle we have
∠DEO' = ∠BEO' (CPCT)
Now ∠AEC=∠DEB
⇒1/2(∠AEC) = 1/2(∠DEB)
⇒∠AEO =∠CEO = ∠DEO'= ∠BEO'
so all 4 ∠'s are equal and bisected by OE and OE'
∴ O,E,O' are collinear