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  • Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.

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Solution
           

Here PQ and PR are tangents and O is the center of circle.
   Let us join OQ and OR.
  Here \angle OQP= \angle ORP= 90^{\circ} 
(\because tangent from exterior point is perpendicular to the radius through the point of contact)
In \bigtriangleupPQO and\bigtriangleupPRO 
OQ= OR   (Radius of circle)
OP= OP         (Common side)
Hence, \bigtriangleup PQO\cong \bigtriangleup PRO      [RHS interior]
Hence, \angle RPO= \angle QPO       [By CPCT]                             
Hence O lie on angle bisector of \angle QPR

Hence Proved.

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