Do you have enough information to work out the area of the shaded quadrilateral?
Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.
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.
