Mathematics
In the figure (i) given below, two circles intersect at A, B. From a point P on one of these circles, two line segments PAC and PBD are drawn, intersecting the other circles at C and D respectively. Prove that CD is parallel to the tangent at P.
Circles
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Answer
From figure,
PT is a tangent and PA is chord.
∠APT = ∠ABP (∵ angles in alternate segments are equal.) …(i)
BDCA is a cyclic quadrilateral as all the vertices lie on the circumference of the circle.
In cyclic quadrilateral the exterior angle is equal to the opposite interior angle.
∴ ∠ABP = ∠ACD ….(ii)
From (i) and (ii),
∠APT = ∠ACD
The angles ∠APT and ∠ACD are alternate angles, but since they are equal,
Hence, proved that CD || PT.
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