X is a point on the tangent at the point A lies on a circle with centre O.A secant drawn from a point X intersects the circle at the points Y and Z, If P is the mid-point of YZ, prove that XAPO or XAOP is a cyclic quadrilateral. .

Explanation

XT is a tangent at A of the circle with centre at O. A secant XZ through X intersects the circel at the point Y and Z.P is the mid-point of YZ.
To prove : XAPOor XAOP is a cyclic quadrilateral.
Proof : YZ is a chord of the circle with centre at O and P is the mid-point of YZ `therefore OP bot YZ.`
`therefore angle OPZ=90^(@)`
Again XT is a tangent at A of the circle with centre at O.
`therefore OA bot XT, therefore angle OAX =90^(@)........(2)`
Now, from (1) and (2) we get,
`angle OPX+ angle OAX =90^(@) +90^(@) =180^(@)`
`therefore` Two opposite angles of the quadrilateral XAOP are supplementary.
`therefore XAOP` is a cyclic quadrilateral. (Proved)