Define a metric $d$ such that $d(x,y)=1$ if $x \not= y$ and $d(x,y)=0$ if $x=y.$ We should also check that $d$ satisfies the metric conditions:
i) $d(x,y) \geq 0$ for all $x$ and $y.$
ii) $d(x,y)=0$ iff $x=y.$ This is true by definition.
iii) $d(x,y)=d(y,x)$ which is obvious.
iv) $d(x,z) \leq d(x,y)+d(y,z)$ holds again by definition.
Therefore $d$ is a metric.
Alright so here is one way of going about it (if I understood you correctly). So let $Z = (z_1,z_2)$ be the point lying on the line through $A,B$ such that the line passing through $C,Z$ is perpendicular to the line through $A,B$. You can find this point as follows.
We can find that $d(A,C) = \sqrt{104}$, $d(C,D) = \sqrt{(z_1-12)^2 + (z_2-10)^2}$, and $d(A,D) = \sqrt{(z_1-10)^2 + (z_2-20)^2}$.
So Pythagorean's theorem gives us the equation:
$$(d(C,D))^2 + (d(A,D))^2 = 104.$$
But we also know that $(z_1,z_2)$ is a solution to the equation of the line through $A,B$, namely the line $y= -\frac{13}{25}x + \frac{630}{25}$.
So now you have two equations and two unknowns. Give that a try and see if you can find $Z$. Once you have $Z$ you should be able to figure out whatever else you were wondering about.
Best Answer
Given only this information, you generally can't know the specific point. That third point will be at an intersection of two circles--one centered around each of the other two points. In particular, let's call your first two points $a,b$ and the point you're looking for $c.$ Let $r_{ab}$ be the distance from point $a$ to point $b$, and let $r_{ac},r_{bc}$ be the desired respective distances from $a,b$ to $c$. What we're dealing with, then, is circles around $a$ and $b$ of respective radii $r_{ac}$ and $r_{bc}$.
Now, if two of $r_{ab},r_{ac},r_{bc}$ add up to the third, then there is exactly one point of intersection of the two circles--and that is our point $c$. If two of $r_{ab},r_{ac},r_{bc}$ add up to some number that is less than the third, then the circles don't intersect at all, and there is no such point $c$. Otherwise, there will be two points of intersection of the circles, and $c$ will be one of those, but (as I said) there's no way to know which, given only this information.