Circle properties and circle theorems

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

What is the same and what is different about these circle questions? What connections can you make?

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?