Let $G$ be a 3-connected planar graph and $z$ a Tutte embedding. A Tutte embedding is defined as the mapping $z:V(G)\to \mathbb{R}^2$ such that i.) there is a face $F$ of the graph $G$ such that $z$ maps the vertices of $F$ to the corners of a strictly convex polygon so that every edge of the face joins consecutive corners of the polygon ii.) every vertex not in $F$ lies at the centre of gravity of its neighbours. The source material is essentially pages 125--127 presented here.
Consider then the proof the lemma 15.3.3 from the page 127 
Then, consider the following picture: 
The drawn graph is certainly 3 connected and planar and the red and blue lines are also possible cuts for the half-space. Suppose that we include everything below the two lines to half-spaces $H_1$ and $H_2$, respectively.
Question: What I don't seem to understand are the assumptions of the author. In that in the drawn picture, the vertex maximizing the distance to the boundary of the half-space is the vertex 0. However the triangle $0-1-3$ is certainly a strictly convex polygon, so I don't see why the vertex $b$ the author refers to has to be outside $F$. Also, should the vertex $a$ be assumed to be different from $b$? Because if $a = b = 0$ and the face $F$ happened to be the polygon $5-3-4-6$, and we were considering the half-space defined by the red line, then $a$ is not necessarily contained in $F$.