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.

Two chords intersect at a point on a circle and the diameter through this point bisects the angle between the chords. Prove that the chords have the same length.

Two chords intersect at a point on a circle and diam¬eter through this point bisects the angle between the chords. Prove that the chords have the same length.

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.

In the picture, the tangent to the circumcircle of a regular pentagon through a vertex is shown. Calculate the angle which the tangent makes with the two sides of the pentagon through the point of contact.

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.

Prove that the tangents drawn to a circle at the two ends of a diameter are parallel.

A chord and the diameter through one of its ends are drawnin a circle. A chord of the same inclination is drawn on the other side of the diameter. Prove that the chords are of the same length.

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 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. .

The gradient at any point (x,y) of a curve is 3x^2-12 and the curve through the point (2,-7). Find the equation of the tangent at the point (2,-7).