Let $A$, $B$ two points with $distance(A, B)= 2d >0$. Let $m=mid(A, B)$. That is, $distance(A,m)=distance(B,m)=d$.
Define $L$ to be the line that passes through $m$ and which is perpendicular with $[A,B]$.
Let $P$ be the half-plan defined by $L$ and which contains $B$.
Are the following claims true (edited)?
Claim 1: Given any point $C \in P$, given any point $x \in [C, m]$, it holds that: $$distance(A, C)-distance(A, x) \geq distance(B, C) - distance(B, x)$$
Claim 2: Let $S=distance(C,X)$. Assume $S>0$. Is it true that
$$distance(A,C)−distance(A,x)≥(distance(B,C)−distance(B,x)) + f(S)$$ with $f(S)>0$. For example $f(S)=α.S$ with alpha>0 a constant.