A circle has centre O and angle POR = angle QOR. Construct tangents
at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q
lie inside, or on, or outside this circle?

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.

Two tangents are drawn to the other circle from the centres of a
pair of circles. What can you say about the chords cut off by these
tangents. Be patient - this problem may be slow to load.

Some(?) of the Parts

Stage: 4 Challenge Level:

A circle touches the lines $OA$ extended, $OB$ extended and $AB$ where $OA$ and $OB$ are perpendicular.

Show that the diameter of the circle is equal to the perimeter of the triangle.