Prove that the perpendicular from the centre of a circle to a chord, bisects the chord.

4

(a)

AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Prove that  EB = ED and AE = CE.

4

(a)

In the given figure, ABCD is a cyclic quadrilateral whose diagonals intersect at P. If \(\angle DBC=70°\) and \(\angle BAC=30°\) , find \(\angle BCD\).
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4

(a)

ABCD is a cyclic quadrilateral with AD || BC Prove that AB = DC.

4

(a)

Prove that the opposite angles of an isosceles trapezium are supplementary.

4

(a)

In figure, a diameter AB of a circle bisects a chord PQ. If AQ || PB, prove that the chord PQ is also a diameter of the circle.

4

(a)

ABC is a triangle and P is a point on the side BC such that AB = AP. If AP produced meets the circumcircle of \(\Delta ABC\) at Q, prove that CP = CQ.

4

(a)

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.

4

(a)

ln figure, equal chords AB and CD intersect each other at Q at right angle. P and R are the midpoints of AB and CD respectively. Show that OPQR is a square.

4

(a)

Prove that the quadrilateral formed by internal angle bisectors of any quadrilateral is cylic.

4

(a)