Let $(M,g)$ be a complete Riemannian manifold with non-positive curvature. Assume that any two points are connected by a geodesic segment.

Pick three distinct points $o$, $a$ and $b$. Let $L_a(t)$ and $L_b(t)$ be the geodesic segments joining $o$ and $a$, and $o$ and $b$, parametrised with unit speed.

For any $t$, let $m(t)$ be the midpoint between $L_a(t)$ and $L_b(t)$.

Is $m(t)$ a geodesic segment?

Remarks:

1. I think we can assume that $M$ is two dimensional.

2. The answer in the case of the Euclidean plane is affirmative.

Let $(M,g)$ be a complete simply connected Riemannian manifold with non-positive curvature. Because of the Hopf-Rinow theorem, any two points are connected by a geodesic segment.

Pick three distinct points $o$, $a$ and $b$. Let $L_a(t)$ and $L_b(t)$ be the geodesic segments joining $o$ and $a$, and $o$ and $b$, parametrised with unit speed.

For any $t$, let $m(t)$ be the midpoint between $L_a(t)$ and $L_b(t)$.

Is $m(t)$ a geodesic segment?

Remarks:

1. I think we can assume that $M$ is two dimensional.

2. The answer in the case of the Euclidean plane is affirmative.