Geodesic triangles in trees are tripods

Subdivide at the at most three interior-edge points among $a,b,c$. This preserves connectivity, edge lengths and absence of cycles: contracting the finitely many new degree-two vertices in a cycle would produce a cycle in the original graph. Remove closed subwalks from any finite path until a simple path remains. Two distinct simple paths with the same endpoints would have a first divergence and a first subsequent reunion, and the intervening arcs form a cycle. Hence there is exactly one finite simple path between any two of the specified points.

Every finite edge route reduces to that simple path by deleting closed subwalks, so its length is at least the simple-path length. Consequently the latter attains the path distance and, parametrized by length, is isometric on every subinterval. It is also the only geodesic image: if a path leaves this simple path through an edge, it must return through the same attaching point (otherwise the finite routes to the two attaching points form a cycle). A geodesic cannot revisit a point. An interior point of any edge of the simple path separates its endpoints, so every continuous joining path must traverse that edge; thus no alternative geodesic can skip part of the simple path.

The intersection of the finite paths $[a,b]$ and $[a,c]$ is an initial path $[a,m]$. Indeed if a point belongs to both, their subpaths from $a$ to that point coincide by uniqueness. Since the two paths are finite unions of closed edge intervals, this common initial path has a last point $m$. After subdivision at $m$, their tails $[m,b]$ and $[m,c]$ meet only at $m$; their concatenation is the simple path from $b$ to $c$. Hence the three pairwise paths have precisely the three legs from $m$, and distances are sums of leg lengths.

Each side is a union of two legs, each also on another side; hence every side lies in the union of the other two. This gives $0$-slimness. If points coincide or $m$ is a vertex of the triangle, the corresponding legs have length zero and the same union identities hold. Only finitely many paths and subdivisions were needed, including in an infinite graph.