I'm trying to prove this statement:
Let $\mathcal{K}$ be a non-degenerate conic section in $\mathbb{R}P^2$, with on it three distinct points $A,B$ and $C$. Let $a$ be the tangent line to $\mathcal{K}$ in $A$, $b$ the tangent in $B$ and $c$ the tangent in $C$. On the line $c$ it holds that: $c\cap AB$ is the harmonic conjugate of $C$ with repect to $c\cap a$ and $c\cap b$.
Because $\mathcal{K}$ is non-degenerate, we can do a projective transformation such that $\mathcal{K} \leftrightarrow \lambda_0x_0^2+\lambda_1x_1^2+\lambda_2x_2^2 = 0$ with $\lambda_0,\lambda_1,\lambda_2\neq 0$. We can also assume $A,B,C$ are projectively independent, otherwise, $\mathcal{K}$ would be degenerate. So we can take $(A,B,C)$ as a basis for our projective coordinates. But here is where I am stuck, I think the result will follow by some calculation, but I am unsure how to go from the lines $a,b,c$ in cartesian coordinates to the points $c\cap AB, c\cap a$ and $c\cap b$ in projective coordinates.
This is proven in Article 178 of Milne's Cross Ratio Geometry. It's a synthetic proof that adds the line passing through $C$ and $a\cap b$ and then does some cross ratio chasing. This should be a pretty stable URL, but here's the excerpt (the names of points are obviously different).