A theorem for cubic-A generalization of Carnot theorem I found a theorem for cubic, the theorem is a generalization of the Carnot theorem for conic. I'm an electrical engineer, not a mathematician. I don't know how to prove this result.
Let ABC be a triangle, Let three points $A_1, A_2, A_3$ lie on $BC$, three points $B_1, B_2, B_3$ lie on $CA$, let three points $C_1, C_2, C_3$ lie on $AB$. There is a cubic passing through the nine points $A_1, A_2, A_3, B_1, B_2, B_3, C_1, C_2, C_3$ if only if:
$$\frac{\overline{A_1B}}{\overline{A_1C}}.\frac{\overline{A_2B}}{\overline{A_2C}}.\frac{\overline{A_3B}}{\overline{A_3C}}.
\frac{\overline{B_1C}}{\overline{B_1A}}.\frac{\overline{B_2C}}{\overline{B_2A}}.\frac{\overline{B_3C}}{\overline{B_3A}}.
\frac{\overline{C_1A}}{\overline{C_1B}}.\frac{\overline{C_2A}}{\overline{C_2B}}.\frac{\overline{C_3A}}{\overline{C_3B}}=1$$


Let $S$ be a hypersurface of degree $d$ of equation $\phi(x) = 0$ in some $k^n$, $a$ a point not on $S$. Consider a line $a +  t v$ through $a$ that intersects $S$ in $n$ points given by the equation in $t$
$$\phi(a + t v ) = 0$$
Now expanding the LHS wr to the powers of $t$ we get
$$\phi(a) + \cdots  + \phi_d(v) t^d =0$$
(where $\phi_d$ is the $d$-homogenous part of $\phi$)). Therefore, the product of the roots of the above equation is ( by Viete)
$$(-1)^d\frac{\phi(a)}{\phi_d(v)}$$
Therefore: if $a$,  are $b$ points not on $S$ and the line $ab$ intersects $S$ in $d$ points $c_1$, $\ldots$, $c_d$ then
$$\frac{ac_1 \cdot ac_2 \cdots ac_d}{b c_1\cdot bc_2 \cdots bc_d} = \frac{\phi(a)}{\phi(b)}$$
Consequence ( Carnot's theorem)
Let $a_1$, $\ldots$, $a_n$ points not on $S$. For every $i$ consider the intersection of the line $a_i$, $a_{i+1}$ with $S$ as $c_{i1}$, $\ldots$, $c_{id}$. Then we have
$$\prod_{i=1}^n \frac{a_i c_{i1} \cdot a_i c_{i2} \cdots \cdot a_i c_{id}}{a_{i+1} c_{i1}\cdot a_{i+1} c_{i2}  \cdots \cdot a_{i+1} c_{id}}=1$$
Indeed, the product equals
$$\prod_{i=1}^n \frac{\phi(a_i)}{\phi(a_{i+1})} = 1$$
Notes: the theorem about the product of chords determined by an algebraic curve is due to Newton ( it generalizes the power of a point)
The theorem of Carnot follows from the theorem of Newton, and generalizes results like Menelaus theorem.
