Inside a circle, a point A is taken. Find the geometric locus of the points of intersection of the tangents to the circle, drawn through the ends of all possible chords containing point $A$.

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Let $O$ be the center of the circle, $R$ its radius, $M$ the point of intersection of the tangents drawn through the ends of a chord containing point $A$, and $P$ the midpoint of this chord. Then $O P \cdot O M=R^{2}$ and $O P=O A \cos \varphi$, where $\varphi=\angle A O P$. Therefore, $A M^{2}=O M^{2}+O A^{2}-2 O M \cdot O A \cos \varphi=O M^{2}+O A^{2}-2 R^{2}$, which means that the value $O M^{2}-A M^{2}=2 R^{2}-O A^{2}$ is constant. Consequently, all points $M$ lie on a line perpendicular to $O A$ (see problem 7.6).