We have $n$ points in the plane such that they are not all collinear. We call a line $\ell$ a 'good' line if we can divide those $n$ points in two sets $A,B$ such that the sum of the distances of all points in $A$ to $\ell$ is equal to the sum of the distances of all points in $B$ to $\ell$. Prove that there exist infinitely many points in the plane such that for each of them we have at least $n+1$ good lines passing through them.
Problem
Source: Iran TST 2011 - Day 3 - Problem 3
Tags: combinatorics proposed, combinatorics
MohammadP
14.05.2011 22:12
Hint :
assume that points are $P_1, ..., P_n$
define : $X = w_1P_1 + w_2P_2 + ... + w_nP_n$
where $|w_i| = |w_j|$ and $w_1+w_2+...+w_n = 1$
prove that EVERY line passing through $X$ is good !
shoki
18.05.2011 10:30
another solution is considering G the centroid of all points and the fact that every line passing thru it is good.then u can prove that after deleting some lines from the plane all other points are good points (that is n+1 good lines pass thru each of them). that means also that we can prove $\mu( \mbox{good points})=1$ !
Hooksway
19.10.2011 16:30
MohammadP wrote: Hint :
assume that points are $P_1, ..., P_n$
define : $X = w_1P_1 + w_2P_2 + ... + w_nP_n$
where $|w_i| = |w_j|$ and $w_1+w_2+...+w_n = 1$
prove that EVERY line passing through $X$ is good !
Hello.Are $w_i$s real or complex?Can you explain it in detail？
P-H-David-Clarence
28.07.2016 12:21
shoki wrote: another solution is considering G the centroid of all points and the fact that every line passing thru it is good.then u can prove that after deleting some lines from the plane all other points are good points (that is n+1 good lines pass thru each of them). that means also that we can prove $\mu( \mbox{good points})=1$ ! what you said was toltally wrong. And for Mohammad P,the point of the question is to show that there are at least n+1such X,can anybody explain why?
mathmaths
31.07.2016 18:02
MohammadP wrote: Hint :
assume that points are $P_1, ..., P_n$
define : $X = w_1P_1 + w_2P_2 + ... + w_nP_n$
where $|w_i| = |w_j|$ and $w_1+w_2+...+w_n = 1$
prove that EVERY line passing through $X$ is good !
Could some mathlinkers explain this hint to me?It seems quite unclear.