The picture on the right shows two circles centred on the same point and a line intersecting them.Prove that the parts of the line between the circles on either side are equal.

The picutre on the right shows two circles centred on the same point and the line intersecting them.Prove that the parts of the line between the circles on either side are equal.

In the picture on the right, the angles between the radii and the chords are equal.Prove that the chords are of the same length.

The picture shows two circles with the same centre and two triangles formed by joining the centre to the points of intersection of the circles with two radii of the larger circle. prove that these triangles are similar .

The picture shows the tangents at two points on a circle and the radii through the points of contact. Prove that the tangents have the same length.

In the picture below, the top vertex of a triangle is joined to the midpoint of the opposite side and then the point dividing this line in the ratio 2:1 is joined to the other two vertices. Prove that the areas of all three triangles in the picture on the right are equal to a third of the area of the whole triangle

In the picture below, two lines are drawn from the top vertex of a triangle to the bottom side: Prove that the ratio in which these lines divide the length of the bottom side is equal to the ratio of the area of the three smaller triangles in the picture.

In the picture below, the top vertex of a triangle is joined to the midpont of the opposite side and then the point dividing this line in the ratio 2:1 is joined to the other two vertices: Prove that the areas of all three triangles in the picture on the are equal to a third of the area of the whole triangle.

The picture below shows two circles, each passing through the center of the other. Calculate the area of the region common to both.

In the picture below, two vertices of a parallelogram are joined to the mid points of two sides. Prove that these lines divides the diagonal in the picture into three equal parts. .

At two ends of a horizontal line, angles of equal size are drawn, and two points on the slanted lines are joined. Prove that the parts of the horizontal line and parts of slanted lines are in the same ratio.