B1. Calculate the coordinates of point $C$, which is equidistant from points $A(-6,1)$ and $B(2,-3)$ and lies on the line $y=3x-6$. Write the equation of the set of all points that are equidistant from points $A$ and $B$.
B1.

1. way

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-20.jpg?height=57&width=1739&top_left_y=2270&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-20.jpg?height=69&width=1739&top_left_y=2324&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-20.jpg?height=58&width=1731&top_left_y=2381&top_left_x=174)

Transforming the equality $v$ into the equation $(x+6)^{2}+(3 x-7)^{2}=(x-2)^{2}+(3 x-3)^{2} \ldots \ldots \ldots . .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-20.jpg?height=52&width=1727&top_left_y=2487&top_left_x=176)

Calculating the midpoint of the segment $A B$ is $S\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)=S(-2,-1) \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . . .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-20.jpg?height=63&width=1725&top_left_y=2598&top_left_x=177)

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-20.jpg?height=60&width=1736&top_left_y=2648&top_left_x=177)

## 2. way

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-21.jpg?height=75&width=1744&top_left_y=271&top_left_x=173)

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-21.jpg?height=68&width=1739&top_left_y=337&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-21.jpg?height=77&width=1742&top_left_y=390&top_left_x=174)

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-21.jpg?height=57&width=1739&top_left_y=454&top_left_x=176)

Determining that point $C$ lies at the intersection of the bisector $y=2 x+3$ and the line $y=3 x+6.1^{*}$ point

Solving the system of equations .................................................................................................................................

Writing the point $C(9,21)$......................................................................................................