A Simple Proof of the Sylvester–Gallai Theorem
The Sylvester–Gallai theorem states that any finite set of points in the Euclidean plane, not all collinear, has a line passing through exactly two of them. This article presents a concise proof by mathematician Leroy Milton Kelly, using the concept of a point-line pair with minimal distance. Assuming a line contains more than two points leads to a contradiction via similar triangles, proving the theorem elegantly.
If we draw a connecting line 𝓂 that passes through P and C, and draw the perpendicular from B to B′ on 𝓂, then BB′ will be shorter than PP′ (because PP′C and BB′C are similar triangles).
- hyperhello
> Every finite set of points in the Euclidean plane that is not collinear has a line that passes through exactly two of the points.
I can't make out the point here (no pun). Of course a line can pass through any two points. It could pass through three if those points were collinear but the statement says they're not. So what is the new fact?
- pratikdeoghare
> According to a strengthening of the theorem, every finite point set (not all on one line) has at least a linear number of ordinary lines. An algorithm can find an ordinary line in a set of n points in time O(n log n). [1]
There are many such lines (think convex hull) and they are easy to find.
This makes it hard to appreciate the theorem.
You keep thinking oh whats the big deal.
[1] https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theor...
- frotaur
In the proof, it claims that "At least two of these must fall on the same side of P′, the perpendicular projection of P on ℓ.".
However, this is not true as it is possible that P'=B. However it seems the proof still goes through (at least as depicted in the image, haven't thought hard about the general case).
- Nail2680
I might be too stupid to understand why this is interesting and useful. If it helps I am a working physicist, and a lot of pure math is lost on me. I think I followed this, but I don't know why one would care or this would be interesting.
- emil-lp
Futility closet is fantastic!