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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.

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Solution
Construction : Join AO and OS
  {O}'D and{O}'B
      
In \bigtriangleup E{O}'B and \bigtriangleup E{O}'D
{O}'D= {O}'B   [Radius are equal]
{O}'E= {O}'E  [Common side]
ED = EB           [Tangent drawn from an external point to the line circle  are equal to length]
AE{O}'B\cong E{O}'D  [By SSS congruence criterion]
\Rightarrow \; \angle {O}'ED= \angle {O}'EB\, \cdots \left ( i \right )
i.e.,      {O}'E  is bisector of\angle BED 
Similarly OE is bisector of \angle AEC 
In quadrilateral DEB{O}'
\angle {O}'DE= \angle {O}'BE= 90^{\circ}
\Rightarrow \, \angle {O}'DE= \angle {O}'BE= 180^{\circ}
\therefore \angle DEB+\angle D {O}'B= 180^{\circ}\, \cdots \left ( ii \right ) [\mathbb{Q}DEB{O}' is cyclic quadrilateral]
\angle AED+\angle DEB= 180^{\circ} [\mathbb{Q}  AB is a straight line]
\angle AED+180^{\circ}-\angle D{O}'B= 180^{\circ} [From equation (ii)]
\angle AED-\angle D{O}'B= 0
\angle AED-\angle D{O}'B\: \: \cdots \left ( iii \right )
Similarly \angle AED= \angle ADC\: \:\cdots \left ( iv \right )     
 From equation (ii) \angle DEB= 180^{\circ}-\angle D{O}'B
 Dividing both side by 2
\frac{\angle DEB}{2}= \frac{180^{\circ}-\angle D{O}'B}{2}
\angle DE{O}'= 90^{\circ}-\frac{1}{2}\angle D{O}'B\: \: \cdots \left ( v \right )
\left [ \because \, \frac{\angle DEB}{2}= \angle DE{O}' \right ]
Similarly \angle AEC= 180^{\circ}-\angle AOC
Dividing both side by 2
\frac{1}{2}\angle AEC= 90^{\circ}-\frac{\angle AOC}{2}
\angle AEO= 90^{\circ}-\frac{1}{2}\angle AOC\; \cdots \left ( vi \right )
\left [ \because \frac{1}{2}\angle AEC= \angle AEO \right ]
Now \angle AEO+\angle AED+\angle DE{O}'
= 90-\frac{1}{2}\angle AOC+\angle AED+90^{\circ}-\frac{1}{2}\angle D{O}'B
= \angle AED+180^{\circ}-\frac{1}{2}\left ( \angle D{O}'B+\angle AOC \right )
= \angle AED+180^{\circ}-\frac{1}{2} \left [ \angle AED+\angle AED \right ] [from equation (iii) and (iv)]
= \angle AED+180^{\circ}-\frac{1}{2} \left [ 2\angle AED \right ]
\angle AED+180^{\circ}-\angle AED
= 180^{\circ}
\therefore \: \angle AED+\angle DE{O}'+\angle AEO= 180^{\circ}
So,OEO’ is straight line
\therefore O, E and {O}'  are collinear.
Hence Proved

 

 

 


 

 

           

 

 

 

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