Distance in the Poincare Disk model of hyperbolic geometry I am trying to understand the Poincare Disk model of a hyperbolic geometry and how to measure distances. I found the equation for the distance between two points on the disk as: 
$d^2 = (dx^2 + dy^2) / (1-x^2-y^2)^2$ 
Given two points on the disk, I am assuming that $dx$ is the difference in the euclidean $x$ coordinate of the points, and similar for $dy$. So, what are $x$ and $y$ in the formula? Also, I see some formulas online as: 
$d^2 = 4(dx^2 + dy^2) / (1-x^2-y^2)^2$
I am not sure which one is correct.
Assuming that $X$ and $Y$ are coordinates of one of the points, I have tried a few examples, but can't get the math to work out. For instance, if $A = (0,0)$, $B = (0,.5)$, $C = (0,.75)$, then what are $d(A,B)$, $d(B,C)$, and $d(A,C)$? The value $d(A,B) + d(B,C)$ should equal $d(A,C)$, since they are on the same line, but I can't get this to work out. I get distances of $.666$, $.571$, and $1.714$ respectively.
MathWorld - Hyperbolic Metric

The point is $dx$ and $dy$ in the formula (the one with the 4 in is right btw) don't represent the difference in the $x$ and $y$ co-ordinates, but rather the 'infinitesimal' distance at the point $(x,y)$, so to actually find the distance between two points we have to do some integration.
So the idea is that at the point $(x,y)$ in Euclidean co-ordinates, the length squared, $ds^2$, of an infinitesimally small line is the sum of the infinitesimally small projections of that line onto the $x$ and $y$ axes ($dx^2$ and $dy^2$) multiplied by a scaling factor which depends on $x$ and $y$ ($\frac{4}{(1-x^2-y^2)^2}$).
