To find the point of intersection among the three spheres, we need to solve the system of equations formed by the equations of the spheres.

Let's denote the coordinates of the point of intersection as (x, y, z).

The equation of the sphere with center (1, 1, 0) and radius 2 units is:
(x - 1)^2 + (y - 1)^2 + z^2 = 2^2

The equation of the sphere with center (3, 4, 0) and radius 2 units is:
(x - 3)^2 + (y - 4)^2 + z^2 = 2^2

The equation of the sphere with center (5, 1, 2) and radius 2 units is:
(x - 5)^2 + (y - 1)^2 + (z - 2)^2 = 2^2

We can simplify the equations by expanding them:

Equation 1: x^2 - 2x + 1 + y^2 - 2y + 1 + z^2 = 4
Equation 2: x^2 - 6x + 9 + y^2 - 8y + 16 + z^2 = 4
Equation 3: x^2 - 10x + 25 + y^2 - 2y + 1 + z^2 - 4z + 4 = 4

Simplifying further, we get:

Equation 1: x^2 - 2x + y^2 - 2y + z^2 = 2
Equation 2: x^2 - 6x + y^2 - 8y + z^2 = -21
Equation 3: x^2 - 10x + y^2 - 2y + z^2 - 4z + 4 = 0

We can subtract Equation 1 from Equation 2 to eliminate x^2, y^2, and z^2:

Equation 2 - Equation 1: -4x - 6y = -23

We can subtract Equation 1 from Equation 3 in a similar manner:

Equation 3 - Equation 1: -8x + 4 - 6y - 4z + 4 = -2
Simplifying further: -8x - 6y - 4z = -6

Now, we have a system of two linear equations with three variables:

-4x - 6y = -23
-8x - 6y - 4z = -6

We can solve this system to find the values of x, y, and z.

Multiplying the first equation by -2, we get:

8x + 12y = 46

Now, we can add this equation to the second equation:

(8x + 12y) + (-8x - 6y - 4z) = 46 - 6
Simplifying further: 6y - 4z = 40

From the equation 2, we can solve for z:

-8x - 6y - 4z = -6
-4z = -6 + 8x + 6y
4z = 6 - 8x - 6y
z = (6 - 8x - 6y) / 4
z = (3 - 4x - 3y) / 2

Now, substituting this expression for z into the equation 6y - 4z = 40:

6y - 4((3 - 4x - 3y) / 2) = 40
6y - 2(3 - 4x - 3y) = 40
6y - 6 + 8x + 6y = 40
12y + 8x - 6 = 40
12y + 8x = 46
6y + 4x = 23

Now, we have a system of two linear equations with two variables:

6y + 4x = 23
6y - 4x = 40

Adding these two equations, we eliminate x:

(6y + 4x) + (6y - 4x) = 23 + 40
12y = 63
y = 63/12
y = 21/4

Substituting this value of y into the first equation, we can find x:

6(21/4) + 4x = 23
63/2 + 4x = 23
4x = 23 - 63/2
4x = 23 - 126/2
4x = 23 - 63
4x = -40
x = -10

Substituting the values of x and y into the equation for z, we can find z:

z = (3 - 4x - 3y) / 2
z = (3 - 4(-10) - 3(21/4)) / 2
z = (3 + 40 + 63/4) / 2
z = (3*4 + 160 + 63)/4 / 2
z = (12 + 160 + 63)/4 / 2
z = 235/4 / 2
z = 235/8

Therefore, the point of intersection among the three spheres is approximately (-10, 21/4, 235/8).

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