Circle properties and circle theorems

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Do points P and Q lie inside, on, or outside this circle?

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

How much of the field can the animals graze?

Keep constructing triangles in the incircle of the previous triangle. What happens?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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?

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Can you find a relationship between the area of the crescents and the area of the triangle?