# 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,
$\angle ABC=\angle ACB......\left(i\right)$

Also,

From (i) and (ii)

$\angle ACB=\angle AQC$

Since APQC is a cyclic quadrilateral,

Now , exterior angle in a triangle is equal to sum of interior opposite angles.
$\angle BCQ+\angle AQC=SRQ....\left(iv\right)$

Using (iv) in (iii);
$\angle APQ+\angle SRQ=180°$

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

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