# On Geodesic Triangles in Hyperbolic Plane

## TL;DR

This paper proves that in a hyperbolic surface, any side of a triangle formed by lifts of a closed geodesic is shorter than the geodesic itself, revealing geometric constraints in hyperbolic geometry.

## Contribution

It establishes a new inequality relating sides of triangles formed by lifts of a closed geodesic in hyperbolic surfaces.

## Key findings

- Any side of a triangle formed by lifts of a closed geodesic is shorter than the geodesic.
- The result applies to orientable hyperbolic surfaces without boundary.
- Provides insight into the geometric structure of hyperbolic surfaces.

## Abstract

Let M be an orientable hyperbolic surface without boundary and let $\gamma$ be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of $\gamma$ in H2 is shorter than $\gamma$.