This paper reports the design of a high-accuracy spatial location estimation method using ultrasound waves by exploiting the fixed geometry of the transmitters. Assuming an isosceles triangle antenna configuration, where three antennas are placed as the vertices of an isosceles triangle, the spatial location problem can be formulated as a non-convex optimization problem whose interior is shown to admit a Riemannian manifold structure. Our investigation of the geometry of the newly introduced manifold (i.e., the manifold of all isosceles triangles in R3) enables the design of highly efficient optimization algorithms. Simulations are presented to compare the performance of the proposed approach with popular methods from the literature. The results suggest that the proposed Riemannian-based methods outperform the state-of-the-art methods. Furthermore, the proposed Riemannian methods require much less computation time compared to popular generic non-convex approaches.