# Prove Sylvester Gallai Theorem using combinatorics

I have come across a question in my book which goes as follows:

Consider a finite set $$S$$ of points in a euclidiean plane such that not all of them are collinear. Prove that there exists a line which passes through exactly two points of $$S$$.

Now this result, known as Sylvester Gallai Theorem (not exactly), can be proven by me using induction but the book demands the proof using combinatorics. In this attempt for same question, the proof seems to be rather complicated.

Please help me. (Also please limit to only high school level math)

Thanks

• Does this answer your question? Proving the existence of a line that passes only through two points Commented Sep 11, 2020 at 6:25
• @ArnaudMortier, with due respect sir, have you noticed that it is the same question I have linked in my post and I was having difficulty understanding the answer for that question. Asking for the explanation of that answer there only would be a completely different question as to what OP asked. Also that question has not been active for a long time now. Commented Sep 11, 2020 at 9:58
• I have, yet you are asking for an easier proof and I don't believe that there is one, as was explained in Meta: this is a well-known problem, with well-known proofs, and the proof from the linked question is about as easy as it gets. Commented Sep 11, 2020 at 17:21
• Perhaps you can articulate where the proof of the theorem is giving you difficulty? Besides saying it "seems to be rather complicated," you've given Readers little insight into why you found it complicated or difficult. Patience in working through its steps may benefit you in more ways than one. Commented Sep 16, 2020 at 1:57