Let $S$ be a $2$-dimensional sphere endowed with a flat metric with $3$ conical singularities of positive curvature. Typically, $S$ is a metric space you get when you glue two copies of the same triangle along its boundary.

I have two questions about simple (not self-intersecting, avoiding singular points), totally geodesic paths joining two singular points.

1) Is there a finite number of such paths ?

2) If it is the case, is there an algorithm to compute this number ? If not, how does this number grow with the length ?