Paths in a Triangular Park

In a park, there are 3 straight paths that form a triangle (there are no additional paths). The entrances to the park are at the midpoints of the paths, and a lamp hangs at each vertex of the triangle. From each entrance, the shortest walking distance along the park's paths to the lamp at the opposite vertex was measured. It turned out that 2 out of the 3 distances are equal to each other. Is the triangle necessarily isosceles?