Distance in Poincaré disk from origin to a point given Let $C$ circle $x^2+y^2=1$ find the distance (Poincaré disk) from $O=(0,0)$ to $(x,y)$
The distance in Poincaré is $d=ln(AB,PQ)$ where AB are a segment of the curve and P and Q are points in the limits of Poincaré disk. Then $A=(0,0)$ and $B=(x,y)$ but I dont know the values of P and Q. I try to use the circunference formula, but I have only two points (A and B)  and I need three. Please give me clues, to solve this problem. Thank you.

See, metric entry in the Poincaré Disk Model :
$$|u| = \sqrt{x^2 + y^2}$$
$$\delta(u, v) = 2 * \frac{|u-v]^2}{(1 - |u|^2)(1 - |v|^2)}$$
In your case, $v = O$ and $|v| = 0$ so :
$$\delta(u, O) = 2 * \frac{|u|^2}{(1 - |u|^2)}$$
$$argcosh(1 + \delta(u, O))$$
