10. Three points $A, B$, and $C$ are located on the same line ($B$ is between $A$ and $C$). Take an arbitrary circle centered at $B$ and denote by $M$ the point of intersection of the tangents drawn from $A$ and $C$ to this circle. Find the geometric locus of points $M$ such that the points of tangency of $A M$ and $C M$ with the circle lie inside the segments $A M$ and $C M$.
10. Since $M B$ is the bisector of angle $A M C$, then $\frac{|A M|}{|M C|}=$ $=\frac{|A B|}{|B C|}$. Therefore, the bisector of the external angle relative to angle $A M C$ intersects line $A C$ at a constant point $K: \frac{|A K|}{|K C|}=$ $=\frac{|A B|}{|B C|}$, and the set of points $M$ is an arc of a circle constructed on $B K$ as its diameter, enclosed between the lines perpendicular to segment $A C$ and passing through points $A$ and $C$.

![](https://cdn.mathpix.com/cropped/2024_05_21_35c23d544adcc44dca41g-115.jpg?height=396&width=402&top_left_y=1359&top_left_x=134)

Fig. 15