A regular tetrahedron has equal edge lengths. The given vertices are \((0, 0, 0)\), \((1, 0, 0)\), \((0, 1, 0)\). The side length is 1.

["Understanding the Regular Tetrahedron with Given Vertices", "A regular tetrahedron is a fundamental three-dimensional geometric figure characterized by four equilateral triangular faces, six equal edges, and four vertices. Defining a regular tetrahedron starts with ensuring all edges are of equal length—a property that brings mathematical precision and symmetry to its structure.", "One way to construct a regular tetrahedron is by specifying a set of four carefully selected points in 3D space. In this article, we explore a regular tetrahedron with three of its vertices given as ((0, 0, 0)), ((1, 0, 0)), and ((0, 1, 0)), confirming that each edge between these points has length 1, consistent with the definition of a regular tetrahedron.", "### Defining the Edge Length", "We are given the points:\n- ( A = (0, 0, 0) )\n- ( B = (1, 0, 0) )\n- ( C = (0, 1, 0) )\n- ( D = (x, y, z) ), the unknown fourth vertex", "Since the tetrahedron is regular, all edges must have equal length. The side length is confirmed to be 1 between ( A, B ), ( A, C ), and ( B, C ):", "- Distance ( AB = \sqrt{(1-0)^2 + (0-0)^2 + (0-0)^2} = 1 )\n- Distance ( AC = \sqrt{(0-0)^2 + (1-0)^2 + (0-0)^2} = 1 )\n- Distance ( BC = \sqrt{(0-1)^2 + (1-0)^2 + 0^2} = \sqrt{1 + 1} = \sqrt{2} )", "Wait—this reveals a key insight: the triangle ( ABC ) has side lengths ( AB = 1 ), ( AC = 1 ), and ( BC = \sqrt{2} ), which does not form an equilateral triangle, let alone the base of a regular tetrahedron. This inconsistency means these three points cannot define the faces of a regular tetrahedron with unit edge length unless the fourth vertex ( D ) adjusts accordingly.", "### Correcting for Equilateral Base", "To build a regular tetrahedron with edge length 1, a proper setup must ensure all six edges are equal, not just some. While placing three points with pairwise distance 1 is possible, ensuring equilateral triangles in a plane such that a fourth point lies equidistant (at distance 1) requires geometric alignment.", "For points ( A = (0, 0, 0) ), ( B = (1, 0, 0) ), and ( C = (0.5, \frac{\sqrt{3}}{2}, 0) ), we obtain an equilateral triangle in the ( xy )-plane with side length 1. But the given vertices use ( C = (0, 1, 0) ), introducing a deviation.", "Instead, suppose we accept the three vertices as given: ((0,0,0)), ((1,0,0)), and ((0,1,0)). These form two edges of length 1, but ( BC = \sqrt{2} ), violating the equilateral condition. Therefore, these specific points cannot define a regular tetrahedron with equal edge lengths of 1 unless we reinterpret or restructure the configuration.", "### The Role of the Fourth Vertex in a Regular Tetrahedron", "For a regular tetrahedron with edge length 1, all vertices must satisfy pairwise distance 1. Given ( A, B, C ) with ( AB = AC = 1 ), but ( BC = \sqrt{2} ), the triangle is isosceles but not equilateral. Thus, no fourth point ( D ) can simultaneously make ( AD = BD = CD = 1 ) and preserve base triangle ( ABC ) as equilateral.", "Hence, the given setup does not produce a regular tetrahedron with all edges equal to 1. However, this highlights the importance of precise vertex selection to maintain regularity.", "### Constructing a Correct Regular Tetrahedron with Edge Length 1", "To form a regular tetrahedron with edge length 1, choose any two points, say ( A = (0, 0, 0) ), ( B = (1, 0, 0) ). Then pick a third point ( C ) in the ( xy )-plane such that ( AC = BC = 1 ). This places ( C ) at ( (0.5, \frac{\sqrt{3}}{2}, 0) ), forming an equilateral triangle.", "The fourth vertex ( D = (0.5, \frac{\sqrt{3}}{6}, \frac{\sqrt{6}}{3}) ) lies above the centroid of triangle ( ABC ), equidistant from all three vertices at distance 1. This point ensures all six edges — ( AB, AC, BC, AD, BD, CD ) — are exactly length 1, fulfilling the definition of a regular tetrahedron.", "### Summary", "- A regular tetrahedron requires all four vertices to be connected by edges of equal length.\n- With vertices ((0,0,0)), ((1,0,0)), and ((0,1,0)), the base triangle has edge lengths (1), (\sqrt{2}), and (1), so it is not equilateral.\n- Therefore, this configuration cannot represent a regular tetrahedron with side length 1.\n- A valid construction begins with an equilateral base in a plane, then places the apex at height (\frac{\sqrt{6}}{3}) above the centroid using coordinates derived from symmetry and distance constraints.", "### Conclusion", "Understanding the geometric constraints of a regular tetrahedron—equal edge lengths, equilateral triangular faces, and spatial symmetry—is essential in 3D modeling, crystallography, and computational geometry. While the given vertices ((0, 0, 0)), ((1, 0, 0)), and ((0, 1, 0)) do not form a regular tetrahedron (due to mismatched edge lengths), exploring their implications deepens appreciation for precise geometric construction. For true regularity, the third vertex must lie in the plane forming an equilateral triangle with side 1, and the apex must be equidistant at unit length from all three.", "Constructing such a tetrahedron exemplifies how mathematical rigor ensures structural integrity across science and engineering.", "---", "Keywords: regular tetrahedron, edge length 1, equilateral triangle, 3D geometry, coordinates in space, spatial symmetry, geometric construction, tetrahedron vertices, crystallography."]