# Thread: Proving that there is no such point on a line

1. ## Proving that there is no such point on a line

Hello,

I'd greatly appreciate any help with the following problem:

Let $\displaystyle l$ be a line and $\displaystyle P$ a point not on the line.
Prove that there is no such point $\displaystyle T_0 \in l$ such that

$\displaystyle (\forall T \in l)$_____$\displaystyle d(P,T) \leq d(P, T_0)$.

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In other words, prove that there is no point $\displaystyle T_0$ on the line, such that the distance from $\displaystyle T_0$ to $\displaystyle P$ is greater than the distances from all the other points of the line to the point $\displaystyle P$.

Just to illustrate the problem, we may take $\displaystyle T_0 \in l$, and in fact $\displaystyle d(T_1,P)\leq d(T_2,P) \leq d(T_3,P) \leq d(T_0,P)$.

However, that is not always true for $\displaystyle T_0$, for there is a point $\displaystyle T_4$ such that $\displaystyle d(T_4,P) \nleq d(T_0,P)$:

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I have tried to prove this problem by supposing that the opposite is true and then arriving at contradiction, and tried to use the triangle inequality $\displaystyle (d(A,B) \leq d(A,C) + d(C,B))$ somewhere on the way, but so far none of my attempts have been successful.

Thanks a lot!

2. ## my thinking

drop a perpendicular from P on the line (let point of intersection be A).
(this is the shortest distance between P and line)
now from P draw a line parallel to the given line and take an arbitrary point B on it.(take initial point of line as P)
now angle APB=90 degrees.
if this drawn line want to touch the other line then angle APB must be smaller then 90 degrees.
suppose we draw a line at the angle which is just smaller than 90 degrees.
let this angle be x degrees.
but since an angle greater than x but smaller than 90 can still be obtained which may be given by
(x+90)/2
hence a point of intersection (T) belonging to l can always be obtained such that PT>PT(o)
or
no point T(o) exist such that PT(o)>PT.

2) let the equation of line be
ax+by+c=0
let coordinates of P be (r,t)
now let the coordinates of perpendicular from P on line be (m,n)
suppose there is a point T(o) whose distance from T is maximum.
Let its coordinates be (M,N)
but another point T can be obtained on l whose coordinates can be given by
(m+M+c/a,n+N) or (m+M,n+N+c/b) whose distance from P i.e PT will be greater than PT(o)
so
there is no point T(o) belonging to l such that PT(o)>PT