I was thinking of an unusual problem in geometry. Given triangle ABC. Define a "rank" as a line-segment drawn from any vertex to any point on the opposite side. To prove: That any two ranks intersect.
Why, do I need this? When math books on geometry prove for example that medians pass though a common point they assume that they intersect first. But if it were not always true then by contropositive their proof is destroyed. Thus, to prove that the medians pass through a common point we must first prove that the medians intersect somewhere.
I believe I have a proof to this from analytic geometry. Construct a triangle on where these points are not collinear. Then you could perhaps use elementary algebra to prove that such a point exists. But if anyone has a geometric proof I would greatly appreciate it.