I was trying to prove that "a hyperbolic circle in $\mathbb{D}^2$ is a Euclidean circle in $\mathbb{D}^2$ and vice-versa" ($\mathbb{D}^2$ is the Poincare's disk) then I think that I need to calculate the distance between two generics points on the Poincare's disk, but I can't find a way to do this without the omega/ideal points, because I need it for this formula:
$$d(z,z_0)=|\ln(z,z_0,z_1,z_2)|=\left|\ln\left(\dfrac{z-z_1}{z-z_2}\cdot\dfrac{z_0-z_2}{z_0-z_1}\right)\right|$$
where $z$ and $z_0$ are the points in $\mathbb{D}^2$ that I want to calculate the distance and $z_1$, $z_2$ are the generic omega/ideal points.
My idea is to calculate this distance to be the radius of the circle $$\sigma=\left\{z\in\mathbb{D}^2:d(z,z_0)=r\right\}$$
where $z_0\in\mathbb{D}^2$ is any center of the circle and $z$ is any point in $\mathbb{D}^2$ and somehow conclude that hyperbolic circle in $\mathbb{D}^2$ is a Euclidean circle in $\mathbb{D}^2$.
So, my questions are: Is there any formula to calculate generic omega/ideal points? Am I on the right way to prove that "a hyperbolic circle in $\mathbb{D}^2$ is a Euclidean circle in $\mathbb{D}^2$ and vice-versa"?
