Past exam of the short options course of the University of Oxford / 2023 / 3 / d

The fundamental theorem of algebra factors $q(x,1)$ into linear factors over $\mathbb{C}$; homogenizing supplies factors of $y$ if the degree is less than three. Thus every nonzero homogeneous cubic is $c{L}_1{L}_2{L}_3$ for linear forms $L_i$, with $c\ne 0$ absorbed into one factor.

For pairwise nonproportional factors, part (c) assigns three distinct tangent lines ${\ell }_i=\mathbb{P}(L_i\cdot \mathrm{span}\{x,y\})$ to $C$. Their pairwise intersections are $[L_1{L}_2]$, $[L_1{L}_3]$, and $[L_2{L}_3]$. There cannot be a common point of all three: a nonzero quadratic cannot be divisible by three pairwise nonproportional linear factors, by unique factorization. Hence the three lines form a triangle circumscribed about $C$.