The general point on xy plane is D(x, y, 0). Consider this point is equidistant to the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).
∴ AD = BD
\(\sqrt{(x-2)^2+(y-0)^2+(0-3)^2}\) = \(\sqrt{(x-0)^2+(y-3)^2+(0-2)^2}\)
Squaring both sides,
(x - 2)2 + (y - 0)2 + (0 - 3)2 = (x - 0)2 + (y - 3)2 + (0 - 2)2
X2 – 4x + 4 + y2 + 9 = X2 + y2 - 6y + 9 + 4
- 4x = -6y ….(1)
Also, AD = CD
\(\sqrt{(x-2)^2+(y-0)^2+(0-3)^2}\) = \(\sqrt{(x-0)^2+(y-0)^2+(0-1)^2}\)
Squaring both sides,
(x - 2)2 + (y - 0)2 + (0 - 3)2 = (x - 0)2 + (y - 0)2 + (0 - 1)2
X2 – 4x + 4 + y2 + 9 = X2 + y2 + 1
-4x = -12 ….(2)
Simultaneously solving equation (1) and (2) we get
X = 3, y = 2.
The point which is equidistant to the points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1) is (3, 2, 0).