Question

In the given figure two circles Intersect at two points A and B. XY is a tangent at point P. Prove that CD is parallel to the tangent XY.​

Answer 1

Construction - Join AB.

Let the tangent at P be = XY

Thus, according to the alternate segment theorem,

∠APX = ∠ABP --- eq 1

Since, ABCD is a cyclic quadrilateral,

Thus, according to the sum of the opposite angles theorem

∠ABD + ∠ACD = 180°

Also ∠ ABD = ∠ABP = 180° ( As it is a linear pair)

Then, ∠ACD = ∠ABP --- eq 2

From equation 1 and 2 -

∠ ACD = ∠ APX

Therefore,  XY || CD ( As alternate angles are equal).