13.3.5 * Given a circle and a point $M$ inside it, consider all possible rectangles $M K T P$, with vertices $K$ and $P$ on the circle. Find the locus of point $T$.
Let $M(a, b)$, the equation of the circle be $x^{2}+y^{2}=r^{2}$, and let $P\left(x_{1}, y_{1}\right), K\left(x_{2}, y_{2}\right), T(x, y)$. Since $M K T P$ is a rectangle, we have
$$
\left\{\begin{array}{l}
\frac{x+a}{2}=\frac{x_{1}+x_{2}}{2}, \\
\frac{y+b}{2}=\frac{y_{1}+y_{2}}{2}, \\
(x-a)^{2}+(y-b)^{2}=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2} .
\end{array}\right.
$$
Squaring and adding (1) and (2), noting that $x_{i}^{2}+y_{i}^{2}=r^{2}(i=1,2)$, we get
$$
(x+a)^{2}+(y+b)^{2}=2 r^{2}+2 x_{1} x_{2}+2 y_{1} y_{2},
$$

From (3), we have
$$
(x-a)^{2}+(y-b)^{2}=2 r^{2}-2 x_{1} x_{2}+2 y_{1} y_{2},
$$
Adding (4) and (5) yields
$$
x^{2}+y^{2}=2 r^{2}-\left(a^{2}+b^{2}\right) .
$$

Therefore, the locus of point $T$ is a circle centered at the origin with radius $\sqrt{2 r^{2}-\left(a^{2}+b^{2}\right)}$.