Question

Two circles intersect at A and B. P is any point on the circumference of one of the circles. PA and PB are joined and produced to meet the circumference of the other circle at C and D respectively. Prove that the tangent at P is parallel to CD.

Answer 1

Given : Two circles intersect at A and B. P is any point on the circumference of the smaller circles. PA and PB are joined and produced to meet the circumference of the larger circle at C and D respectively.

Prove that: the tangent at P is parallel to CD. i.e

Construction : Draw aTangent   through Point ' P '. Join and

Proof :

In the above figure is the tangent of the smaller circle is the chord.

   is the angle formed out of the the tangent and the chord of the

Now is Inscribed in arc PBA

........... (BY ALTERNATE SEGMENT THEOREM : The alternate segment theorem states that an angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. In the above diagram, the alternate segment theorem tells us that are equal. ) ................. ( 1 )

Now Points A , B , D and C lie on the circumference of the Larger Circle.   They are Concyclic

is a Cyclic Quadrilateral and is its exterior angle.

............ (Exterior angle property of Cyclic Quadrilateral : The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.) ................... ( 2 )

From 1 and 2   

By Alternate angle test of Parallel lines,

i.e.

Thus Proved

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