Example 3. As shown in Figure 3, from a point $Q(2,0)$ outside the circle $x^{2}+y^{2}=1$, a secant is drawn intersecting the circle at points $A$ and $B$. Find the equation of the locus of the midpoint of chord $\mathrm{AB}$.
Connect $\mathrm{OP}$,
then $O P \perp A B$, that is, $\angle \mathrm{OPQ}$ is always a right angle.

Since $OQ$ is a fixed line segment, by the locus theorem of plane geometry:
point $P$ lies on the circle with $O Q$ as its diameter.
Since $\mathrm{P}$ is the midpoint of chord $\mathrm{AB}$, point $\mathrm{P}$ must be inside the circle $x^{2}+y^{2}=1$, therefore, the locus equation of point $P$ is $(x-1)^{2}+y^{2}=1\left(x^{2}+y^{2}<1\right)$.