The circumcenter is the point where all the perpendicular bisectors intersect.
The circumcenter is taken as P. This means that the distance from P to each vertex is equal.
EP = FP = GP
For EP and FP,
EP^2 = FP^2
The coordinates of vertex E is (4, 4)
The coordinate of vertex F is (4, 2)
It means that
(x - 4)^2 + (y - 4)^2 = (x - 4)^2 + (y - 2)^2
(x - 4)(x - 4) + (y - 4)(y - 4) = (x - 4)(x - 4) + (y - 2)(y - 2)
x^2 - 4x - 4x + 16 + y^2 - 4y - 4y + 16 = x^2 - 4x - 4x + 16 + y^2 - 2y - 2y + 4
x^2 - 8x + 16 + y^2 - 8y + 16 = x^2 - 8x + 16 + y^2 - 4y + 4
x^2 - x^2 + y^2 - y^2 - 8x + 8x - 8y + 4y = 4 + 16 - 16 - 16
- 4y = - 12
y = - 12/4
y = 3
Also,
FP^2 = GP^2
(x - 4)^2 + (y - 2)^2 = (x - 8)^2 + (y - 2)^2
(x - 4)(x - 4) + (y - 2)(y - 2) = (x - 8)(x - 8) + (y - 2)(y - 2)
x^2 - 4x - 4x + 16 + y^2 - 2y - 2y + 4 = x^2 - 8x - 8x + 64 + y^2 - 2y - 2y + 4
x^2 - 8x + 16 + y^2 - 4y + 4 = x^2 - 16x + 64 + y^2 - 4y + 4
x^2 - x^2 + y^2 - y^2 - 8x + 16x - 4y + 4y = 64 + 4 - 16 - 4
8x = 48
x = 48/8
x = 6
Therefore, the circumcenter, P of the triangle has the coordinates, (6, 3)
