AB and AC are two equal chords of a circle and AP, AQ are two other chords which intersect BC in S and R respectively. Prove that PQRS is a cyclic quadrilateral.

To prove PQRS is cyclic, it suffices to prove that a pair of opposite angles is supplementary.Join BC, BP and CQ as shown

In triangle ABC, since chords AB and AC are equal,

ABC =AQC (angles in the same segment).....(ii)

From (i) and (ii)


Since APQC is a cyclic quadrilateral,

APQ+ACQ=180°APQ+ACB+BCQ =180°APQ+AQC+BCQ =180° ( asACB=AQC.. as proved above)....(iii)

Now , exterior angle in a triangle is equal to sum of interior opposite angles.

Using (iv) in (iii);

These are opposite angles of quadrilateral PQRS. So, PQRS is a cyclic quadrilateral


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