The common tangents AB and CD to two circles with centres O and O’ intersect at E between their centres. Prove that the points O,E and O’ are collinear

Dear student
Given : AB and CD are common tangents to the circles with centre O and O' intersecting at E.
We know that tangents drawn from an exterior point are equally inclined to the segment joining the centre to that point
=> angle AEO = angle CEO
or
EO is the angle bisector of angle AEC ........(1)
and angle BEO' = angle DEO'
or
EO' is the angle bisector of angle BED ........(2)
but angle AEC = angle BED .............(3)
from (1), (2) and (3) we can conclude that Eo' coincides with EO
or
E, O', O are coillinear.
Regards
Arun (askIITians forum expert)