Triangle Centres. Determine the Coordinates

Question

On a map three towns are located at the coordinates A(5, 1), B(-2, 0) and C(4, 8). The township wants to build a new fire station that is the same distance from each town. Determine the coordinates where they should build the new fire station. (Hint: The circumcentre is the same distance from each vertex.)

Answer 1

Let P(x,y) be the point that is equidistant from the 3 towns.

Using the 2D distance formula:

√((x-5)² + (y-1)²) = √((x+2)² + (y-0)²) = √((x-4)² + (y-8)²)

Squaring all 3 expressions:

(x-5)² + (y-1)² = (x+2)² + (y-0)² = (x-4)² + (y-8)²

The squared terms drop out, and collecting like terms we have

-10x -2y + 26 = 4x + 4 = -8x -16y + 80

Or, as 3 equations:

-10x -2y + 26 = 4x + 4 Eqn A

4x + 4 = -8x -16y + 80 Eqn B

-10x -2y + 26 = -8x -16y + 80 Eqn C

Collecting like terms:

14x + 2y = 22 A'

12x + 16y = 76 B'

-2x + 14y = 54 C'

Multiply A' by 7, and subtract C':

100x = 100

So

x = 1

Substituting x = 1 in Eqn A, we get y = 4

So the equidistant point is at (1,4).

To verify, plug (1,4) into the 3 distance expressions (the distance is 5).