In Fig. 9.26, $A$, $B$, $C$ and $D$ are four points on a circle. $AC$ and $BD$ intersect at a point $E$ such that $\angle BEC = 130^{\circ}$ and $\angle ECD = 20^{\circ}$. Find $\angle BAC$.

$ABC$ and $ADC$ are two right triangles with common hypotenuse $AC$. Prove that $\angle CAD = \angle CBD$.

Two circles intersect at two points $B$ and $C$. Through $B$, two line segments $ABD$ and $PBQ$ are drawn to intersect the circles at $A$, $D$ and $P$, $Q$ respectively (see Fig. 9.27). Prove that $\angle ACP = \angle QCD$.

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?

Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

In Fig. 9.25, $\angle ABC = 69^{\circ}$, $\angle ACB = 31^{\circ}$, find $\angle BDC$.

Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

If a line intersects two concentric circles (circles with the same centre) with centre $O$ at $A$, $B$, $C$ and $D$, prove that $AB = CD$ (see Fig. 9.12).

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

In Fig. 9.23, $A, B$ and $C$ are three points on a circle with centre $O$ such that $\angle BOC = 30^{\circ}$ and $\angle AOB = 60^{\circ}$. If $D$ is a point on the circle other than the arc $ABC$, find $\angle ADC$.

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

$ABCD$ is a cyclic quadrilateral whose diagonals intersect at a point $E$. If $\angle DBC = 70^{\circ}$, $\angle BAC$ is $30^{\circ}$, find $\angle BCD$. Further, if $AB = BC$, find $\angle ECD$.

In Fig. 9.24, $\angle PQR = 100^{\circ}$, where $P$, $Q$ and $R$ are points on a circle with centre $O$. Find $\angle OPR$.

Prove that a cyclic parallelogram is a rectangle.