A(0,5) B(0,2) C(4,2)
In Euclidean geometry the three points given are the vertices of a right-angled triangle. Find the three angles of the hyperbolic triangle with vertices A,B,C. Find the hyperbolic lengths of the three sides.
I know the sum of the interior angles of a hyperbolic triangle are less than 180 but I have no idea where to go from here.
To find the distances, you can plug the points into the Poincare half-plane model distance formula.
To find the angles, one way to do this is to find the lines between the points. In the upper half-plane model, the geodesics are vertical lines and semicircles which are perpendicular to the $x$-axis. Therefore, you can find that $A$ and $B$ are on the line $x=0$. The points $B$ and $C$ are on the semicircle $(x-2)^2+y^2=8$. The points $A$ and $C$ are on the semicircle with center $(t,0)$ where $t$ solves $t^2+5^2=(4-t)^2+2^2$ (that is, the point on the $x$ axis that is the same distance two $A$ and $C$). Finally, you can take the angles between these curves as hyperbolic space is locally Euclidean.