Pythagoras' theorem

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Can you find the distance between the two trees using the information given?

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

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.