A triangle in the Poincare disc model

Question

Suppose that we have a triangle in the Poincare disc model such that the internal angels are all equal. Then Does it imply that the lengths of sides are all equal?

By length of a side, I mean the length of geodesics between two vertices of the triangle.

Answer 1

Let $ABC$ be the triangle, with $a = |BC|$ and $b = |AC|$. The angles opposite these sides are $\alpha =\angle CAB= \angle ABC$.

Now assume the contrary $a < b$. Choose point P on AC such that $|PC|= |BC|= a$ and let $\omega$ be the angle of $CPB$. Thus $\angle CPB = \angle CBP = \omega$.

We know that $\alpha + (\alpha-\omega)+\angle APB < \pi$ and moreover $\omega+\angle APB = \pi$.

Hence, we can conclude that $\alpha<\omega$ and a contradiction.

Answer 2

the triangle is equilateral since if two angles are congruent then the corresponding sides are congruent in any triangle in a Hilbert's space geometry. See Hartshorne: Geometry