Q. 103.9( 15 Votes )

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.


Answer :

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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