##### In a figure the common tangents, AB and CD to two circles with centers O and O’ intersect at E. Prove that the points O, E and O’ are collinear.

Given : AB and CD are two tangents with centers O and O' intersect at E .

To Prove : O, E and O' are collinear.

Construction : Join AO, OC O'D and O'B

In AOE and EOC

OA = OC [radii of same circle]

OE = OE [common]

AE = EC [Tangents drawn from an external point to a circle are equal]

AOE EOC [By Side Side Side Criterion]

AEO = CEO [Corresponding parts of congruent triangles are equal ]

AEC = AEO + CEO = AEO + AEO = 2AEO [1]

Now As CD is a straight line

AED + AEC = 180° [linear pair]

2AEO = 180 - AED [From 1]

[2]

Now, In O'ED and O'EB

O'B = O'D [radii of same circle]

O'E = O'E [common]

EB = ED [Tangents drawn from an external point to a circle are equal]

O'ED O'EB [By Side Side Side Criterion]

O'EB = O'ED [Corresponding parts of congruent triangles are equal ]

DEB = O'EB + O'ED = O'ED + O'ED = 2O'ED [3]

Now as AB is a straight line

AED + DEB = 180 [Linear Pair]

2O'ED = 180 - AED [From 3]

[4]

Now,

So O, E and O' lies on same line [By the converse of linear pair]

Hence Proved.

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