The solution to this will be a circle in 3 dimensions.

Since Q - P = [2, 2, 2] and the midpoint between P and Q is M = [3, 4, 5], and 2[1, 1, 1] = [2, 2, 2],the points on the circle are in the plane x + y + z = 3 + 4 + 5 = 12.

MR = √(PR^{2 }- MP^{2}) = √(5^{2 }- (1^{2}+1^{2}+1^{2}) ) = √22

Thus, our solution is the set of points on x + y + z = 12 that are √22 from [3, 4, 5]

For example, one definition of the points, as z = 12 - x - y, is

(x - 3)^{2} + (y - 4)^{2} + (12 - x - y - 5)^{2} = 22

z = 12 - x - y

or

(x - 3)^{2} + (y - 4)^{2} + (7 - x - y)^{2} = 22

z = 12 - x - y