Prove the radius of the incircle of an equilateral triangle is half the radius of its circumcircle.

What sort of a quadrilateral is formed by the tangents at the ends of two perpendicular diameters of a circle?

The picture shows the tangents at two points on a circle and the radii through the points of contact.

Prove that the line joining the centre and the point where the tangents meet is the perpendicular bisector of the chord joining the points of contact.

In the picture, the sides of the large triangle are tangents of the circumcircle of the smaller triangle, through its vertices.

Calculate the angles of the smaller triangle.

In the picture, $PQ, RS, TU$ are tangents to the circumcircle of $\Delta ABC$.

Sort out the equal angles in the picture.

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?

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