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?
Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.
In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.
Try to make sense of each method one line at a time. The following mathematical topics may be useful:
Pythagoras' theorem
Cosines of angles greater than $90^{\circ}$
Distance between two points on a coordinate grid
Trigonometry in right-angled triangles