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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.

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