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📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \boxed{-\frac{7}{3}}

📌 ---Question: A bioinformatician is analyzing a dataset of DNA sequences, each represented as a point in a 3D space where each coordinate corresponds to the frequency of a nucleotide (A, C, G, T). If three such sequences correspond to the points $(1, 2, 3), (4, 5, 6),$ and $(7, 8, 9)$, find the coordinates of the fourth point that would form a regular tetrahedron with integer coordinates, ensuring all edges are of equal length.

📌 Solution: To find the fourth vertex of a regular tetrahedron with the given three vertices $A = (1, 2, 3)$, $B = (4, 5, 6)$, and $C = (7, 8, 9)$, we first compute the pairwise distances between the given points:

📌 $AB = \sqrt{(4-1)^2 + (5-2)^2 + (6-3)^2} = \sqrt{9 + 9 + 9} = \sqrt{27} = 3\sqrt{3}$

📌 $AC = \sqrt{(7-1)^2 + (8-2)^2 + (9-3)^2} = \sqrt{36 + 36 + 36} = \sqrt{108} = 6\sqrt{3}$

📌 $BC = \sqrt{(7-4)^2 + (8-5)^2 + (9-6)^2} = \sqrt{9 + 9 + 9} = \sqrt{27} = 3\sqrt{3}$

📌 We observe that $AB = BC = 3\sqrt{3}$, but $AC = 6\sqrt{3}$, so the three points are not equidistant. Thus, they cannot form a face of a regular tetrahedron. Therefore, the given points cannot be vertices of a regular tetrahedron with integer coordinates.

📌 However, if we reinterpret the question as seeking a fourth point $D = (x, y, z)$ such that all six edges $AB, AC, AD, BC, BD, CD$ are equal and the coordinates are integers, we proceed by solving the system:

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