Show point is the harmonic conjugate of intersections of tangents on non-degenerate conic section

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.