Approximate Steiner tree with exact angles II

The root is at the bottom. There are 2^3+1 terminals, including the root. The approximate Steiner tree is in blue. The angles are exactly, 120, 120-epsilon, 120+epsilon. The Steiner minimal tree for the same topology, in red, is constructed using the Melzak algorithm. There is a slider with which you can change epsilon. Now there is a construction of a broken path [math]p_0p_1p'_2p''_5q_1[/math] which has the same length as the approximate tree. The straight-line path [math]p_0q_1[/math] is the length of the Steiner minimal tree. Equal angles have been indicated. If you look inside one of the small circles, say the circumcircle of triangle [math]p_8p_9q_4[/math], then you'll see that the angle [math]p_4q_4p_8[/math] has to be at most 30 degrees for the Melzak construction to work. Another requirement is that this circle should not swallow the Steiner point [math]s_2[/math], but I don't yet know how to specify that.