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 shows a triangle formed by three tangents to a circle. . Calculate the length of each tangent from the corner of the triangle to the point of contact.

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.

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

For the curve y=4x^3-2x^5 , find all points at which the tangent passes through the origin.

A regular pentagon is drawn with vertices on a circle of radiús 15 centimetres. Calculate the length of its sides.

For the curve y=4 x^3-2 x^5 , find all the points at which the tangent passes through the origin.

In the picture below, the lines parallel to each side of the blue triangle through the opposite vertex are drawn to make the big triangle. How many triangles in the picture have the same area as that of the blue 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 lines parallel to each side of the blue triangle through the opposite vertex are drawn to make the big triangle. How many triangles in the picture have the same area as that of the blue triangle? How many of them have the same measures of the blue triangle?