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. (in $, cm$)

Let the vertex of an angle $\angle ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that $\angle ABC$ is equal to half the difference of the angles subtended by the chords $DE$ and $AC$ at the centre.

In the figure,$AB$ is a diameter of the circle,and $CD$ is a chord equal to the radius of the circle. $AC$ and $BD$ when extended intersect at a point $E$. Prove that $\angle AEB = 60^{\circ}$.

In the figure,$\angle ABC = 69^{\circ}$ and $\angle ACB = 31^{\circ}$. Find $\angle BDC$. (in $^{\circ}$)

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 figure). 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 lies on the third side.