Example 6 In the Cartesian coordinate system $x O y$, given three points $A\left(0, \frac{4}{3}\right), B(-1,0), C(1,0)$. The distance from point $P$ to line $B C$ is the geometric mean of the distances from the point to lines $A B$ and $A C$. Find the equation of the locus of point $P$.
(2004 National High School Mathematics Competition)
The equations of the lines $AB$, $AC$, and $BC$ are $y=\frac{4}{3} \cdot(x+1)$, $y=-\frac{4}{3}(x-1)$, and $y=0$, respectively. The distances from point $P(x, y)$ to $AB$, $AC$, and $BC$ are $d_{1}=\frac{1}{5}|4 x-3 y+4|$, $d_{2}=\frac{1}{5} \cdot|4 x+3 y-4|$, and $d_{3}=|y|$, respectively, and $d_{1} d_{2} d_{3} \neq 0$. According to the problem, we have: $\left|16 x^{2}-(3 y-4)^{2}\right|=25 y^{2}$, which means $16 x^{2}-(3 y-4)^{2}+25 y^{2}=0$, or $16. x^{2}-(3 y-4)^{2}-25 y^{2}=0$. Simplifying, we get the trajectory equation of point $P$ as the circle $S: 2 x^{2}+2 y^{2}+3 y-2=0$ and the hyperbola $T: 8 x^{2}-17 y^{2}+12 y-8=0$, excluding points $B$ and $C$.