Question

B
P
Two chords of a circle with centre O intersect inside it
at P. A and B are the middle points of the chords. Prove
that O, A, P and B are concyclic points.
OA​

Answer 1

Given : Two chords of a circle with centre O intersect inside it  at P.

A and B are the middle points of the chords

To Find : Prove  that O, A, P and B are concyclic points.

Solution:

concyclic points. - concyclic points are at the same distance from the center of the circle.

Hence concyclic points should lie on a circle

=> OAPB will be a cyclic Quadrilateral -  a quadrilateral whose vertices all lie on a single circle.

A Quadrilateral is a cyclic Quadrilateral  if sum of opposite two angles = 180°

A & B are the middle points of chord

=> OA and OB are perpendicular to chords

P is the intersection of chords

=> OA ⊥  AP   and  OB ⊥ BP

∠A + ∠B = 90° + 90° = 180°

Hence sum of opposite two angles = 180°

=> OAPB is a cyclic quadrilateral

Hence points O , A , P & B are concyclic points

QED

Hence Proved

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