Question: A museum curator catalogs a 3D artifact with three vertices of a regular octahedron at $ (1, 0, 0) $, $ (-1, 0, 0) $, and $ (0, 1, 0) $. Find the coordinates of the fourth vertex, given all coordinates are integers.

["Title: Discovering the Missing Vertex: The Fourth Octahedron Coordinate Solved", "When a museum curator catalogs a 3D artifact, precision and geometric insight are essential—especially when dealing with highly symmetric shapes like the regular octahedron. A fascinating challenge arises when three vertices are known: $ A(1, 0, 0) $, $ B(-1, 0, 0) $, and $ C(0, 1, 0) $. What’s the integer-coordinate fourth vertex of this regular octahedron?", "### Understanding the Regular Octahedron", "A regular octahedron is one of the five Platonic solids, featuring eight equilateral triangular faces, six vertices, and twelve edges. It has two apex vertices aligned along a central axis, with four equilateral triangular faces forming a square middle layer when visualized that way.", "Crucially, all edges are of equal length, and every pair of vertices at maximal symmetry distances defines the shape.", "Given vertices:\n- $ A = (1, 0, 0) $\n- $ B = (-1, 0, 0) $\n- $ C = (0, 1, 0) $", "We seek the fourth vertex $ D = (x, y, z) $, such that all edges from $ D $ to $ A, B, C $ are equal in length to the edges defining the rest of the octahedron.", "### Step 1: Determine Edge Length", "First, compute the distance between $ A $ and $ B $:", "$$\nAB = \sqrt{(1 - (-1))^2 + (0 - 0)^2 + (0 - 0)^2} = \sqrt{4} = 2\n$$", "Now distance $ AC $:", "$$\nAC = \sqrt{(1 - 0)^2 + (0 - 1)^2 + (0 - 0)^2} = \sqrt{1 + 1} = \sqrt{2}\n$$", "Wait—this is inconsistent: $ AB = 2 $, $ AC = \sqrt{2} $. But in a regular octahedron, all edges must be equal.", "Hence, not all three given points lie on the same face or are mutually adjacent. Let’s reassess their relative positions.", "### Step 2: Verify Triangular Face Possibility", "For the points to form part of the same regular octahedron with equal edge lengths, each pairwise distance must equal the edge length $ s $. But $ AC = \sqrt{2} $, $ AB = 2 $, so they cannot be adjacent.", "This indicates the three points do not form a face of the octahedron—they are not mutually connected by edges.", "Instead, consider the octahedron’s known structure: it has two poles (e.g., $ (0, 0, \pm h) $) and four equatorial vertices in a symmetric square.", "But our points $ (1,0,0), (-1,0,0), (0,1,0) $ lie in the $ xy $-plane, suggesting they may lie on the equatorial square—provided symmetry holds.", "In a regular octahedron with equatorial symmetry, if equatorial vertices lie in the $ z = 0 $ plane, their arrangement is often $ (a, a, 0), (-a, a, 0), \dots $, but here $ (1,0,0) $ and $ (0,1,0) $ suggest orthogonal axes.", "Let’s suppose the full regular octahedron centered at the origin has vertices along the coordinate axes. The standard regular octahedron centered at the origin with vertex-aligned symmetry has vertices at:", "$$\n(\pm a, 0, 0),\ (0, \pm a, 0),\ (0, 0, \pm a)\n$$", "But then all edges have length $ \sqrt{2}a $, and each face is an equilateral triangle.", "But in our case, distances are mismatched.", "Alternatively, note that the three given points: $ (1,0,0), (-1,0,0), (0,1,0) $, lie on the $ xy $-plane. For a regular octahedron inscribed in a sphere centered at the origin, symmetry suggests the two missing equatorial vertices might be $ (0,0, z) $ and $ (0,0,-z) $, or orthogonal poles.", "But we need to find the fourth vertex, assuming the three given ones are part of a face or diagonal.", "Wait—reconsider: in a regular octahedron, each vertex connects to four others. The three points $ (1,0,0), (-1,0,0), (0,1,0) $ cannot all be adjacent to each other, since $ d((1,0,0), (-1,0,0)) = 2 $, $ d((1,0,0), (0,1,0)) = \sqrt{2} $, which are unequal.", "Therefore, they cannot be mutually adjacent.", "But suppose they lie such that $ (1,0,0) $ and $ (-1,0,0) $ are opposite vertices (antipodal). In a regular octahedron, antipodal vertices are separated by diameter, so distance $ 2s $, where $ s $ is edge length.", "Let $ s $ be the edge length. The distance between antipodal vertices is $ 2s $.", "So if $ AB = 2 $, then $ 2s = 2 \Rightarrow s = 1 $. So edge length is 1.", "Then each edge must be of length 1.", "Now verify: is $ AC = \sqrt{2} $? But $ \sqrt{2} <br/>\ne 1 $. So $ AC $ is not an edge.", "Similarly, $ BC = \sqrt{( -1 - 0 )^2 + (0 - 1)^2} = \sqrt{1 + 1} = \sqrt{2} <br/>\ne 1 $. So $ BC $ is not an edge.", "Thus, $ (1,0,0), (-1,0,0), (0,1,0) $ are all at distance $ \sqrt{2} $ or 2 from each other—not all edges—but possibly non-adjacent vertices.", "But in a regular octahedron, every vertex connects to four others. There are only six vertices. Given three, we seek the fourth such that all edges from it to known ones (if connected) are length $ s = 1 $, or possibly longer if not adjacent.", "But the problem states: “three vertices of a regular octahedron”—so they must all be vertices, but not necessarily forming a face.", "Let’s suppose the octahedron has two poles $ (0,0,a) $, $ (0,0,-a) $, and four equatorial vertices.", "Let’s test whether the three given points could lie on the equator.", "Suppose the equatorial square has vertices at:\n- $ (1,1,0) $—too large,\n- or better: symmetric points like $ (1,0,0), (-1,0,0), (0,1,0), (0,-1,0) $? But distances:\n - $ (1,0,0) $ to $ (0,1,0) $: $ \sqrt{2} $\n - $ (1,0,0) $ to $ (-1,0,0) $: $ 2 $\n - $ (1,0,0) $ to $ (0,-1,0) $: $ \sqrt{2} $", "So if equatorial points are $ (1,0,0), (-"]