Three friends are planning to visit each other. To optimize travel time, they want the meeting place to be equidistant from the three different cities they live in. The cities are located at A(-16, -1), B(1, 6), and C(1, -18). What are the coordinates where the meeting should take place?

The center of the circle lies on the intersection of the perpendicular bisectors of the sides.

Since one side lies on the line x=1, its perpendicular bisector is the line y = -6

Now find the midpoint of one other side, and see where its bisector intersects y = -6.

To find the coordinates where the meeting should take place, we need to find the point that is equidistant from each of the three cities.

Let's denote the meeting point coordinates as (x, y). The distance between two points (x1, y1) and (x2, y2) is calculated using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

We want the distance from the meeting point to city A to be equal to the distance from the meeting point to city B, and also equal to the distance from the meeting point to city C. This can be expressed as three equations:

√((x - (-16))^2 + (y - (-1))^2) = √((x - 1)^2 + (y - 6)^2)
√((x - (-16))^2 + (y - (-1))^2) = √((x - 1)^2 + (y - (-18))^2)
√((x - 1)^2 + (y - 6)^2) = √((x - 1)^2 + (y - (-18))^2)

Simplifying each of these equations, we get:

(x + 16)^2 + (y + 1)^2 = (x - 1)^2 + (y - 6)^2
(x + 16)^2 + (y + 1)^2 = (x - 1)^2 + (y + 18)^2
(x - 1)^2 + (y - 6)^2 = (x - 1)^2 + (y + 18)^2

Expanding the equations and simplifying, we have:

x^2 + 32x + 256 + y^2 + 2y + 1 = x^2 - 2x + 1 + y^2 - 12y + 36
x^2 + 32x + 256 + y^2 + 2y + 1 = x^2 - 2x + 1 + y^2 + 36y + 324
x^2 - 2x + 1 + y^2 - 12y + 36 = x^2 - 2x + 1 + y^2 + 36y + 324

Canceling out like terms, the equations simplify to:

2x + 2y = -258
2y - 12y = 288
-14y = 288

Solving for y, we find:

y = -288 / 14
y = -20.57 (approx.)

Substituting the value of y back into any of the equations, we can solve for x:

2x + 2(-20.57) = -258
2x - 41.14 = -258
2x = -258 + 41.14
2x = -216.86
x = -108.43 (approx.)

Therefore, the meeting point should be approximately at coordinates (-108.43, -20.57).