Twelve synchronized swimmers are forming a circle. the location of those swimmers are (13,-2) (-1,-2) and (6,-9). a 4th swimmer will appear in the center of the circle. where would the center swimmers need to be located?

Let the location of the center (x,y)
let the distance between center and point (13,-2) = d₁
let the distance between center and point (-1,-2) = d₂
let the distance between center and point (6,-9) = d₃

The condition of the circle is d₁ = d₂ = d₃
Making two equations to find x , y

d₁² = d₂²
∴ (x-13)² + (y+2)² = (x+1)² + (y+2)²
∴ (x-13)²  = (x+1)²
∴ x-13 = (x+1) ⇒⇒⇒ positive sign unacceptable
∴ x-13 = -(x+1)
∴ x = 6

d₁² = d₃²
∴ (x-13)² + (y+2)² = (x-6)² + (y+9)²
substituting with x and simplifying the equation
∴ y = -2

∴center = (6,-2)

The center swimmers need to be located at (6,-2)

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