Two circles intersect each other in point A and B . Secants through A and B intersect circles in C, D and M, N as shown in the figure prove that : `CM||DN`

In the adjoining figure, two circles intersect each other at points M and N . Secants drawn from points M and N intersect cirecls at point R, S, P and Q as shown in the figure.

In the adjoining figure, two circles intersect each other at points A and E . Their common secant through E intersects the circle at points B and D . The tangents of the circles at point B and D intersect each other at point C . Prove that squareABCD is cyclic .

Two circles intersect at A and B. From a point P on one of the circles lines PAC and PBD are drawn intersecting the second circle at C and D. Prove that CD is parallel to the tangent at P.

Two circle with centre O and P intersect each other in point C and D. Chord AB of the circle with centre O touches the circle with circle with centre P in Point E. Prove that angleADE+angleBCE= 180^(@)

In the adjoining fig. circles with centres X , Y touch each other at Z . A secant passing through Z meets the circles at A and B respectively. Prove that, radius X A || radius YB. Fill in the blanks and complete the proof.

Two circles C_1a n dC_2 intersect at two distinct points Pa n dQ in a line passing through P meets circles C_1a n dC_2 at Aa n dB , respectively. Let Y be the midpoint of A B ,a n dQ Y meets circles C_1a n dC_2 at Xa n dZ , respectively. Then prove that Y is the midpoint of X Zdot

PA and PB are tangents to a circle S touching it at points A and B. C is a point on S in between A and B as shown in the figure. LCM is a tangent to S intersecting PA and PB in S at points L and M, respectively. Then the perimeter of the triangle PLM depends on o

Two circles intersect each other such that each circle passes through the centre of the other. If the distance between their centres is 12, what is the radius of each circle ?