Two circles intersect at two points A and B. AD and AC are diameters to the two circles (see Fig.10.34). Prove that B lies on the line segment DC.

Two circles intersect at A and and AC,AD are respectively the diameter of the circle . Prove that the points C,B and D are collinear.

Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.

In fig., two circles with centres O, O' touch externally at a point A. A line through A is drawn to intersect these circles in B and C. Prove that the tangents at B and C are parallel.

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

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 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.

A point C lies between two points A and B such that AC = BC, then AC=1/2AB . Point C is called a midpoint of line segment AB. Prove that every line segment has one and only one midpoint.

Two different non-parallel lines meet the circle abs(z)=r . One of them at points a and b and the other which is tangent to the circle at c. Show that the point of intersection of two lines is (2c^(-1)-a^(-1)-b^(-1))/(c^(-2)-a^(-1)b^(-1)) .