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.

In the picture, two circles touch at a point, and the common tangent at this point is drawn.

Prove that this tangent bisects another common tangent of these circles.

In the picture, two circles touch at a point, and the common tangent at this point is drawn.

Prove that the points of contact of these two tangents form the vertices of a right triangle.

In the picture, two circles touch at a point, and the common tangent at this point is drawn.

What is special about the quadrilateral formed by joining the points of contact of the circles?

In the picture below, $AB$ is a diameter and $P$ is a point on $AB$ extended. A tangent from $P$ touches the circle at $Q$. What is the radius of the circle?

In the first picture below, the line joining two points on a circle is extended by $4$ centimetres and a tangent is drawn from this point. Its length is $6$ centimetres, as shown:

The second picture shows the same line extended by $1$ centimetre more and a tangent drawn from this point. What is the length of this tangent?

Draw an equilateral triangle and a semicircle touching its two sides, as in the picture.