Question:** A biologist studying the genetic diversity of plants is analyzing a regular tetrahedron with vertices at \((0, 0, 0)\), \((1, 0, 0)\), \((0, 1, 0)\), and a fourth vertex \((x, y, z)\) where \(x, y, z\) are integers. Find the coordinates of the fourth vertex.

["Title: The Genius Behind the Tetrahedron: Solving the Geometry Puzzle of a Plant DNA Model", "---", "Question: A biologist studying genetic diversity in plants analyzes a regular tetrahedron defined by the points ((0, 0, 0)), ((1, 0, 0)), ((0, 1, 0)), and a fourth vertex ((x, y, z)) with integer coordinates. What are the coordinates of this fourth vertex?", "---", "Answer:\nIn a regular tetrahedron, all six edges are of equal length. Given three vertices:\n(A = (0, 0, 0)),\n(B = (1, 0, 0)),\n(C = (0, 1, 0)),", "we know the edge length between any two of these points is (1), since (AB = BC = CA = 1).", "Let the fourth vertex be (D = (x, y, z)), with (x, y, z \in \mathbb{Z}). For the tetrahedron to be regular, the distance from (D) to each of (A), (B), and (C) must also be exactly 1.", "We set up equations using the distance formula:", "1. Distance (AD = 1):\n[\n\sqrt{x^2 + y^2 + z^2} = 1 \Rightarrow x^2 + y^2 + z^2 = 1\n]", "2. Distance (BD = 1):\n[\n\sqrt{(x-1)^2 + y^2 + z^2} = 1 \Rightarrow (x-1)^2 + y^2 + z^2 = 1\n]", "3. Distance (CD = 1):\n[\n\sqrt{x^2 + (y-1)^2 + z^2} = 1 \Rightarrow x^2 + (y-1)^2 + z^2 = 1\n]", "Now subtract equation (1) from equation (2):\n[\n(x-1)^2 - x^2 = 0 \Rightarrow x^2 - 2x + 1 - x^2 = 0 \Rightarrow -2x + 1 = 0 \Rightarrow x = \frac{1}{2}\n]", "Similarly, subtract equation (1) from (3):\n[\n(y-1)^2 - y^2 = 0 \Rightarrow y^2 - 2y + 1 - y^2 = 0 \Rightarrow -2y + 1 = 0 \Rightarrow y = \frac{1}{2}\n]", "Now substitute (x = \frac{1}{2}), (y = \frac{1}{2}) into equation (1):\n[\n\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + z^2 = 1 \Rightarrow \frac{1}{4} + \frac{1}{4} + z^2 = 1 \Rightarrow z^2 = \frac{1}{2}\n\Rightarrow z = \pm \frac{\sqrt{2}}{2}\n]", "But (z) must be an integer — and (\frac{\sqrt{2}}{2}) is irrational. This implies no such point (D = (x, y, z)) with integer coordinates satisfies all edge lengths equal to 1 if (A, B, C) are at unit distance.", "However, wait: the edge length between (A), (B), and (C) is not 1 — let’s recompute that.", "Compute distance (AB):\n[\nAB = \sqrt{(1-0)^2 + (0-0)^2 + (0-0)^2} = 1\n]\n(AC = \sqrt{(0-0)^2 + (1-0)^2 + 0^2} = 1)\n(BC = \sqrt{(0-1)^2 + (1-0)^2 + 0^2} = \sqrt{1 + 1} = \sqrt{2})", "So triangle (ABC) lies in the (xy)-plane, with (AB = AC = 1), but (BC = \sqrt{2}). Thus, side lengths are not equal, so it is not equilateral — but the full tetrahedron must have all six edges equal in a regular tetrahedron.", "Therefore, for the tetrahedron to be regular, all pairwise distances must be equal. But from above, (AB = 1), (BC = \sqrt{2}), already unequal.", "Hence, the only way a regular tetrahedron can exist with vertices (A), (B), (C) at ((0,0,0)), ((1,0,0)), ((0,1,0)), is if the assumed coordinates are incorrect — or the fourth vertex does not lie in the same plane.", "But the key insight: it is impossible for a regular tetrahedron to have three vertices with pairwise distances (1, 1, \sqrt{2}) — because regular tetrahedron requires all six edges equal.", "So for a regular tetrahedron, all pairwise distances must be equal. But here, (AB = 1), (AC = 1), (BC = \sqrt{2}), so not equal.", "Therefore, the initial assumption must be flawed — unless the biologist discovers the real fourth vertex via symmetry.", "Let’s reframe: Find integer coordinates ((x, y, z)) such that all six distances among the four points are equal — but that’s impossible with two edges of length 1 and one of (\sqrt{2}).", "Alternative interpretation: Perhaps the tetrahedron is regular, but not embedded in the coordinate grid — and the biologist seeks integer-coordinate vertices up to isomorphism.", "But there is a known regular tetrahedron with integer coordinates:\nFor example, one such regular tetrahedron has vertices at:\n((1,1,1)), ((1,-1,-1)), ((-1,1,-1)), ((-1,-1,1)), all with pairwise distances (\sqrt{8} = 2\sqrt{2}).", "But our points (A(0,0,0)), (B(1,0,0)), (C(0,1,0)) are in the plane (z=0), and symmetric about (x+y=z).", "Let us re-solve correctly under the constraint: the tetrahedron is regular, so all edges equal. Let edge length be (s).", "Then:\n(AB^2 = 1^2 = 1)\n(AC^2 = 1^2 = 1)\n(BC^2 = (1-0)^2 + (0-1)^2 + 0^2 = 1 + 1 = 2) → (BC = \sqrt{2})", "So already, two edges differ. Thus, no regular tetrahedron can have these three points unless we reinterpret.", "But wait — perhaps the biologist is studying a geometric model of genetic variation where symmetry implies a fourth vertex such that all edges from (D) to (A,B,C) are equal, and the base triangle is equilateral — but here it is right-angled, not equilateral.", "So instead, reconsider: maybe the biologist assumes the tetrahedron is regular, and despite integer coordinates, finds (D = (x,y,z)) with integer components such that distances match.", "Let’s suppose the edge length is (s). Then:", "From earlier:", "1. (x^2 + y^2 + z^2 = s^2)\n2. ((x-1)^2 + y^2 + z^2 = s^2)\n3. (x^2 + (y-1)^2 + z^2 = s^2)", "Subtract 1 from 2:\n((x-1)^2 - x^2 = 0 \Rightarrow -2x + 1 = 0 \Rightarrow x = 1/2)", "Similarly, subtract 1 from 3: (y = 1/2)", "Then plug into 1:\n((1/2)^2 + (1/2)^2 + z^2 = s^2 \Rightarrow 1/4 + 1/4 + z^2 = s^2 \Rightarrow z^2 = s^2 - 1/2)", "For (z^2) to be rational, (s^2) must be multiple of 1/2, but for (z) to be integer, (z^2) must be integer, so (s^2 \geq 1/2) and (s^2 - 1/2) integer → (s^2 = k + 1/2), not possible for integer (z).", "Thus, no integer solution exists for a regular tetrahedron with vertices at ((0,0,0)), ((1,0,0)), ((0,1,0)), and a fourth with integer coordinates.", "But the biologist observes symmetry — so perhaps the triangle is not in the plane? No, the points are given in plane.", "Wait — unless the tetrahedron is regular, and the three points are not connected by edges of equal length? But in a tetrahedron, every pair of vertices is connected.", "Hence, the only resolution is that the biologist realizes the fourth vertex must be such that all edges from it to (A,B,C) are equal, and those distances equal the longest edge — but given (AB = AC = 1), (BC = \sqrt{2}), the consistent edge length for regular tetrahedron must satisfy equality.", "But since (BC = \sqrt{2} > 1), all edges in regular tet cannot be 1.", "So the biologist concludes: the only way for a regular tetrahedron to have vertices (A,B,C) in the given positions with integer (D) is if the edge length is adjusted and symmetry is respected.", "But known fact: There is no regular tetrahedron with three vertices forming a unit right triangle in the plane.", "However, suppose we scale the model. Let us suppose the biologist instead considers a regular tetrahedron with integer coordinates, and identifies one vertex as ((0,0,0)), another at ((1,1,0)), etc.", "But the question specifies: three points: ((0,0,0)), ((1,0,0)), ((0,1,0)), fourth ((x,y,z)), integers.", "After careful analysis, no such regular tetrahedron exists.", "But the biologist, puzzled, suspects an error — but then remembers: regular tetrahedrons can have vertices at\n[\n(1,1,1),\ (1,-1,-1),\ (-1,1,-1),\ (-1,-1,1)\n]\ndistance between any:\n[\n\sqrt{(1-1)^2 + (1-(-1))^2 + (1-(-1))^2} = \sqrt{0 + 4 + 4} = \sqrt{8} = 2\sqrt{2}\n]", "All edges (2\sqrt{2})."]