Three. (20 points) Let the moving point $P$ be on the line $l_{1}: y=x-4$. Draw two tangents $PA$ and $PB$ from $P$ to the circle $\odot C: x^{2}+y^{2}=1$, where $A$ and $B$ are the points of tangency. Find the equation of the trajectory of the midpoint $M$ of segment $AB$.
Three, let point $P\left(x_{0}, y_{0}\right)$, tangent points $A\left(x_{A}, y_{A}\right), B\left(x_{B}, y_{B}\right)$. Then the equations of the tangent lines $P A$ and $P B$ are
$$
l_{P A}: x_{A} x+y_{A} y=1, l_{B P}: x_{B} x+y_{B} y=1 \text {. }
$$

Since $P$ is the intersection point of the two tangent lines, we have
$$
x_{A} x_{0}+y_{A} y_{0}=1, x_{B} x_{0}+y_{B} y_{0}=1 \text {. }
$$

Thus, the coordinates of points $A$ and $B$ satisfy the equation $x_{0} x+y_{0} y=1$, i.e., $l_{A B}: x_{0} x+y_{0} y=1$.
On the other hand, $l_{o P}: y_{0} x=x_{0} y$.
Let point $M(x, y)$. Then

Also, point $P$ lies on the line $y=x-4$, so
$$
\begin{array}{r}
\frac{y}{x^{2}+y^{2}}=\frac{x}{x^{2}+y^{2}}-4 \\
\Rightarrow x^{2}+y^{2}-\frac{x}{4}+\frac{y}{4}=0 .
\end{array}
$$