The lengths of two parallel chords of a circle are $6\, cm$ and $8\, cm$. If the smaller chord is at a distance of $4\, cm$ from the centre,what is the distance of the other chord from the centre (in $, cm$)?

Bisectors of angles $A, B$ and $C$ of a triangle $ABC$ intersect its circumcircle at $D, E$ and $F$ respectively. Prove that the angles of the triangle $DEF$ are $90^{\circ} - \frac{1}{2}A, 90^{\circ} - \frac{1}{2}B$ and $90^{\circ} - \frac{1}{2}C$.

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.

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.

If two circles intersect at two points,prove that their centres lie on the perpendicular bisector of the common chord.

$A$ 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.