9. (16th Russian Mathematical Olympiad) Given a circle and a point $M$ inside it, consider all possible rectangles $M K T P$, with vertices $K, P$ on the circle. Find the locus of point $T$.

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9. (16th Russian Mathematical Olympiad) Given a circle and a point $M$ inside it, consider all possible rectangles $M K T P$, with vertices $K, P$ on the circle. Find the locus of point $T$.
9. Let the equation of the circle be $x^{2}+y^{2}=r^{2}$, and the points be $M(a, b), K\left(x_{1}, y_{1}\right), P\left(x_{2}, y_{2}\right), T(x, y)$, then $a^{2}+b^{2}<r^{2}, \frac{x+a}{2}=$ $\frac{x_{1}+y_{2}}{2}(1), \frac{y+b}{2}=\frac{y_{1}+y_{2}}{2}(2),(x-a)^{2}+(y-b)^{2}=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}$, squaring (1) and (2) and using $x_{i}^{2}+y_{i}^{2}=$ $r^{2}(i=1,2)$ we get $(x+a)^{2}+(y+b)^{2}=2 r^{2}+2 x_{1} x_{2}+2 y_{1} y_{2}$ (4), adding (3) and (4) yields $x^{2}+y^{2}=2 r^{2}-\left(a^{2}+b^{2}\right)$, hence the required locus is a circle centered at $(0,0)$ with radius $\sqrt{2 r^{2}-\left(a^{2}+b^{2}\right)}$.