Example 1. Let $\mathrm{P}$ be a fixed point inside the circle $\odot \mathrm{O}$, and draw two perpendicular lines through $\mathrm{P}$ intersecting the circle at points $\mathrm{A}$ and $\mathrm{B}$. Find the equation of the locus of the midpoint of chord $\mathrm{AB}$.
As shown in Figure 1, take the line PO as the x-axis, with O as the origin to establish a Cartesian coordinate system. (Set up the coordinate system)

Let the radius of the circle be R, |PO| = a, and the midpoint of chord AB be M(x, y). (Set points)
Connect PM and OM, then |PM| = |MB|.
Also, |OB|^2 = |OM|^2 + |MB|^2,
Therefore, |OB|^2 = |OM|^2 + |PM|^2.
(List equations)
Substitute with the distance formula to get: (x + a)^2 + y^2 + x^2 + y^2 = R^2,
which simplifies to x^2 + y^2 + ax - \frac{R^2 - a^2}{2} = 0, which is the equation of the curve sought.

When finding the equation of a curve, setting up the coordinate system is fundamental. Different coordinate systems can lead to varying levels of complexity in deriving the equation. If the Cartesian coordinate system is set up as shown in Figure 2, the process of deriving the equation becomes more complicated, and the result
\[
\left(x - \frac{a}{2}\right)^2 + \left(y - \frac{b}{2}\right)^2 = \frac{2R^2 - a^2 - b^2}{4} \text{, is not }
\]
as simple as the previous one. When solving problems using the direct method, listing equations is key. It is essential to fully utilize the geometric properties of the conditions, and if necessary, add auxiliary lines. Sometimes, equations can also be listed based on the equal angle relationships.