Circle properties and circle theorems

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

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

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

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.

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

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

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

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?