In Figure, the common tangent, AB and CD to two circles with centres O and O' intersect at E. Prove that the points O, E, O' are collinear.
Solution
Construction : Join AO and OS
and
In and
[Radius are equal]
[Common side]
ED = EB [Tangent drawn from an external point to the line circle are equal to length]
[By SSS congruence criterion]
i.e., is bisector of
Similarly OE is bisector of
In quadrilateral
[ is cyclic quadrilateral]
[ AB is a straight line]
[From equation (ii)]
Similarly
From equation (ii)
Dividing both side by 2
Similarly
Dividing both side by 2
Now
[from equation (iii) and (iv)]
So,OEO’ is straight line
O, E and are collinear.
Hence Proved