Euclidean/Non Euclidean Geometry?

Question

In the Poincare half-plane, let A = i, B = 2i and C = (3 + 4i)/5. Show that triangle ABC is a right triangle. (Which vertex is the right angle at?) Compute the lenghts of its sides. Does the sum of the squares of the legs equal the square of the hypotenuse?

Answer 1

Recall that in the Poincare half plane the straight lines (or sides of the triangle) are either vertical lines, or half-circles with center on the real axis. Thus the segment from A to B is just a piece of the vertical line from i to 2i. The other two "lines" are harder to find, but you can do it by finding the unique circles with center on the real axis containing A and C, or A and B. These circles are (|| || is Euclidean norm):

For A, C: ||z||^2 = 1

For B, C: ||z + 5/2||^2 = 41/4

Since the metric on the Poincare plane is conformal to the Euclidean metric, and the vertical line from A to B meets ||z||^2 = 1 at a right angle in the Euclidean metric, this angle is a right angle in the Poincare plane.

As for calculating the lengths, well that sounds too hard for me.

Answer 2

=2
||A|| = ||B|| = 1

||C||^2 = = (1/25)*(9+16) = 1

This does not appear to be a right triangle.