Task 7. The line $c$ is given by the equation $y=2 x$. Points $A$ and $B$ have coordinates $A(2 ; 2)$ and $B(6 ; 2)$. On the line $c$, find the point $C$ from which the segment $A B$ is seen at the largest angle.
Answer: $(2 ; 4)$.

Solution. The desired point is point $C(2 ; 4)$. Indeed, consider a circle centered at point $P(4 ; 3)$ with radius $P A = P B$. It touches the line $c$ at point $C$. The angle $A C B$ is half the arc $A B$. For any point $C^{\prime}$ on the line $c$, different from point $C$, the angle $A C^{\prime} B$ is half the difference of the arcs $A B$ and $D E$ (see figure), i.e., smaller.

For points $C^{\prime}$ on the line $c$, lying in the other half-plane relative to $A B$, the following reasoning can be made: consider a second circle through $A, B$, tangent to $c$, and call the point of tangency $D$. Then, for the same reason, the angle $A D B$ is greater than or equal to the angle $A C^{\prime} B$. However, the angle $A D B$ is less than $A C B$, since they subtend equal chords, but the radius of the second circle is larger.