The following problem was posed to me by someone who had been wondering about it since his high-school geometry course (many years ago). I had fun with it.
Suppose I have a triangle in mind, but all I tell you about it is the lengths of the three altitudes (in nondecreasing order, say). Have I given you enough information to reconstruct the triangle? If so, give a method for recovering the triangle from this information; if not, give a counterexample (two noncongruent triangles with the same altitude lengths).