Can someone explain to me why $\alpha' < \alpha$ and $\beta'<\beta$ when point $p_l$ is inside the circle? There is suppose to be a way to see this using Thales' theorem.
Also if $p_l$ is inside the circle defined by $p_i,p_j,p_k$ then how does one show that $p_k$ is inside the circle defined by $p_i,p_j,p_l$.