Thanks to last nights University Challenge, I have found a new word for UOAIE: unsportsmanlike.
#2. Two circles of radii r and s intersect at two points such that at each point of intersection the tangents to the circles are mutually perpendicular. Show that the centres of the circles and the points of intersection lie on a third circle, and find the radius of this circle.
So what possible transformations are isometries of the plane? You might think that there were a whole bunch of them: rotations, reflections, and translations. (A reflection followed by a translation is sometimes called a glide reflection.) In fact the picture is simpler than that: it turns out that reflections and translations can be built up from rotations alone! A translation in a certain direction is simply a reflection in two axes perpendicular that direction, while a rotation about a point O is a reflection in two axes through O.Hence any translation is a reflection in two axes perpendicular to the direction of translation whose distance apart is half the distance to be translated.
Hence any rotation is a reflection in two axes through the centre of rotation whose angular separation is half the angle to be rotated through.
5. HAM-SANDWICH THEOREM
It is always possible to slice a three-layered ham sandwich with a single cut of a knife in such a way that each layer of the sandwich is divided into two exactly equal halves by the cut.
The ham-sandwich theorem can be proved using the Borsuk–Ulam theorem.