A circle is a two dimensional object that has constant width;
its height remains constant regardless of the orientation. (In other
words, any two parallel lines, each tangent to the circle is the same
distance apart). Can you construct another two-dimensional figure that
has constant width, but is not a circle? Perhaps surprisingly,
it is possible!

The Reuleaux triangle is probably the best known curve of 'constant
width', and is constructed by drawing circular arcs from each vertex of
an equilateral triangle between the other two vertices. See for
example, Reuleaux
triangle

My friend and colleague from the SA Mathematics Olympiad for many
years, James Ridley, recently (April 2010) wondered whether one could
generalize the Reuleaux triangle to any scalene triangle. To his
surprise (and mine) he found it was possible! Read his short paper at Reuleaux
triangle Generalization which includes a link to a short video
clip showing a dynamic Sketchpad construction of the
generalization.

Unfortunately as often happens, the result is not new, but was
nonetheless an interesting exercise. For more information on Curves of
Constant Width (including this generalization) go to Curves of
Constant Width and scroll the page about half-way down.