In Umemoto's thesis 1 on Dirichlet fundamental domains for Fuchsian groups, Theorem 24 involves assumed angles for subdivided hexagons in the proof (Fig.18 on page 35). However, the rationale behind these angles is unclear.
For instance, I thought, as its reflection $\theta_6$ of bottom left corner of the top triangle should be the top left corner of the internal triangle.
Could someone explain why these angles are chosen and their role in the proof?
Fig.18
The author only goes as follows;
we can find that the angles of the hexagon is like a figure in Fig. 18.
