23. Let $A, B, C$ form an arithmetic sequence, find the equation of the locus of the midpoint of the chord formed by the intersection of the line $A x+B y+C=0$ and the parabola $y=-2 x^{2}$.
23. $\because A 、 B 、 C$ form an arithmetic sequence, $\therefore A-2 B+C=0$, which means for any $A, B, C$, the line $A x+B y+C=0$ passes through the point $P_{0}(1,-2)$. The point $P_{0}(1,-2)$ is also on the parabola $y=-2 x^{2}$. Let the midpoint of the chord formed by the intersection of the line $A x+B y+C=0$ and the parabola $y=-2 x^{2}$ be $P(x, y)$, and let the other intersection point of $P_{0} P$ with the parabola be $P_{1}\left(x_{1}, y_{1}\right)$, then
$$
\left\{\begin{array}{l}
x_{1}=2 x-1, \\
y_{1}=2 y+2 .
\end{array}\right.
$$
$\because P_{1}\left(x_{1}, y_{1}\right)$ is on the parabola, we get $2 y+2=-2(2 x-1)^{2}$, which simplifies to $y+1=-(2 x-1)^{2}$, which is the equation of the trajectory of point $P$.