Proof: Slowly scale (contract) the triangle down to a point. The three corners of the triangle trace the three medians of the triangle. Therefore, the three medians intersect at a point.
Please don't take this harshly -- I think it's a very nice idea. But others have not been honest enough: the idea of this proof is not correct. I think it cleverly hides this, and seems to be possibly correct, but if you try to write down the details, you will not get anywhere with it.
The main problem is that you say to "Slowly scale (contract) the triangle down to a point."
But what point? Scaling a triangle requires that you pick a point to scale it down to, and move the vertices towards that point at a uniform rate.
The only reasonable answer to "what point" is "scale it down to the centroid of the triangle." But then, the proof is circular: in order to give the scaling argument, you already have to know that the three lines all go through the centroid.
Q: Can't I just say to scale down the figure, and not specify what point I'm scaling it down to?
You could, but then it would not be clear what it means for the three corners of the triangle to "trace out" lines during this scaling. In order for them to trace out lines, they must move towards a specified point.
Q: OK. Then can I just pick an arbitrary point to scale it down to?
If you did this, then the lines traced out by the three corners of the triangle would be arbitrary lines, and not necessarily the medians of the triangle.
Q: What if I define the centroid first (say, the average of the three vertices' coordinates), and then scale down to that point?
This will work, but the next step (after defining the centroid) will be to prove that all three medians go through the centroid. And once you have proven that, the scaling argument becomes unnecessary, because you already know that the three medians intersect in a point. The proof is already done.
Q: How could I have seen, before going into this detail, that this proof would not work?
The best "heuristic" reason is that it is not clear, from reading the proof, what facts it uses about the median lines. It seems like the proof could be understood by someone who does not even know what a "median" is. Thus, the proof must be wrong.