I am looking to check my thinking, and for help to make workings rigorous for the following problem.
Let $\mathbf{a},\mathbf{b},\mathbf{c}\in \mathbb{R}^2$, where $\mathbf{a}=(a_1,a_2)^T$ etc. and define
$$f(\mathbf{a},\mathbf{b},\mathbf{c})= \begin{vmatrix} 1 & a_1 & a_2\\ 1 & b_1 & b_2\\ 1 & c_1 & c_2\\ \end{vmatrix}$$
I have shown that $f=0\iff \mathbf{a},\mathbf{b},\mathbf{c}$ colinear in $\mathbb{R}^2$. The next part of the question says:
By giving a geometric interpretation to the condition $f\gt 0$, show that for all $n\geq 3$, there exists $\mathbf{p}_1,...,\mathbf{p}_n\in\mathbb{R}^2$ such that $f(\mathbf{p}_i,\mathbf{p}_j,\mathbf{p}_k)\gt 0$ for all $1\leq i\lt j\lt k \leq n$.
I am concerned I may be overcomplicating this, but this is the workings I have so far:
Note firstly all the points must be distinct, otherwise $f=0$. Clearly the condition means that the points in $\mathbb{R}^2$ are not colinear, and by writing $\mathbf{a}'=(1,a_1,a_2)$ etc. we can write $$ f=(\mathbf{a}'\wedge\mathbf{b}')\,\cdot\,\mathbf{c}' =\lvert\mathbf{a}'\wedge\mathbf{b}'\rvert\lvert\mathbf{c}'\rvert\cos{\theta} $$
where $\theta$ is the angle between $\mathbf{a}'\wedge\mathbf{b}'$ & $\mathbf{c}'$, so we have that the condition holds iff $\theta\in[0,\frac{\pi}{2}]$. Equally, by the cycling property of the scalar triple product, this same condition holds for $\mathbf{b}'\wedge\mathbf{c}'$ & $\mathbf{a}'$, and $\mathbf{c}'\wedge\mathbf{a}'$ & $\mathbf{b}'$.
(*) Since the points in $\mathbb{R}^2$ are not colinear, they must form a triangle and, based on the examples I have sketched and visualised, I believe we must have that the points must appear in the order $\mathbf{a},\mathbf{b},\mathbf{c}$ as we sweep anticlockwise around the triangle's orthocentre.
Then for any $n\geq 3$ we can let the $\mathbf{p}_i$ be the $n$ vertices of a regular $n$-gon in $\mathbb{R}^2$, with the vertices labelled with $i$ increasing anticlockwise.
I may be incorrect with this, and so any hints in the right direction would be great. If not, I would really like some help with making my workings in (*) rigorous. Thank you! Any and all feedback would be appreciated.