Circle properties and circle theorems

Can you find triangles on a 9-point circle? Can you work out their angles?

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

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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?

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

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

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?