Which has the greatest area, a circle or a square inscribed in
an isosceles, right angle triangle? A picture is made by joining
five small quadrilaterals together to make a large
quadrilateral. Is it possible to draw a similar picture if all
the small quadrilaterals are cyclic? The circumcentres of four
triangles are joined to form a quadrilateral. What do you notice
about this quadrilateral as the dynamic image changes? Can you
prove your conjecture? What is the same and what is different
about these circle questions? What connections can you make?
ABCD is a rectangle and P, Q, R and S are moveable points on the
edges dividing the edges in certain ratios. Strangely PQRS is
always a cyclic quadrilateral and you can find the angles. Two
semi-circles (each of radius 1/2) touch each other, and a semi-
circle of radius 1 touches both of them. Find the radius of the
circle which touches all three semi-circles. An equilateral
triangle is sitting on top of a square. What is the radius of
the circle that circumscribes