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.
["Why Points with $AB = BC = 3\sqrt{3}$, $AC = 6\sqrt{3}$ Cannot Form a Face of a Regular Tetrahedron", "In geometry, understanding spatial relationships between points is essential—especially when determining whether a set of points can be vertices of a regular tetrahedron. Consider three points A, B, and C such that $AB = BC = 3\sqrt{3}$ and $AC = 6\sqrt{3}$. While these distances suggest symmetry, they actually violate a key geometric constraint. Let’s explore why these points cannot lie on the face of a regular tetrahedron and, more broadly, why they cannot even represent equidistant vertices on a flat plane with integer coordinates.", "### Analyzing the Distances", "The given distances are:\n- $AB = BC = 3\sqrt{3} \approx 5.196$\n- $AC = 6\sqrt{3} \approx 10.392$", "Observe that $AC = 2 \ imes AB$. This implies triangle $ABC$ is isosceles but not equilateral—the side $AC$ is twice as long as the other two sides. For the points to form a face of a regular tetrahedron, they must be equidistant from each other, forming an equilateral triangle with all sides equal. Since this condition fails, the three points cannot lie on a single face of a regular tetrahedron.", "### Why Equidistance Matters", "A regular tetrahedron is a three-dimensional shape with four equilateral triangular faces, where all six edges are of equal length. Every face is an equilateral triangle with side length equal for all three edges. Thus, for any three vertices of a regular tetrahedron, the distances between each pair must satisfy $AB = BC = CA$ — a strict equidistance condition.", "Here, $AB <br/>\ne AC$ invalidates this requirement. No matter how you rotate or reflect the coordinate system, a planar equilateral triangle with sides $3\sqrt{3}, 3\sqrt{3}, 6\sqrt{3}$ is impossible because such a triangle cannot exist: the longest side would exceed or violate the triangle inequality as expected, but more fundamentally, its geometry contradicts equidistance.", "### The Problem with Integer Coordinates", "Suppose someone claims the three points have integer coordinates — for example, $(x_i, y_j)$ with integer values — and serve as vertices of such a tetrahedron. Integer coordinates imply all coordinate differences are integers. The squared distance between two points with integer coordinates is given by:", "[\nd^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2\n]", "For $AB = 3\sqrt{3}$, squared distance is:\n[\nAB^2 = (3\sqrt{3})^2 = 27\n]", "Similarly, $BC^2 = 27$ and $AC^2 = (6\sqrt{3})^2 = 108$", "Now, check whether three integer-coordinate points can achieve:\n- $AB^2 = BC^2 = 27$\n- $AC^2 = 108$", "This is mathematically possible on the plane, but the geometry forbids such a configuration from being part of a regular tetrahedron because equidistance fails.", "More critically, when attempting to build a 3D regular tetrahedron, all edges between four points must be equal — and planar faces must close under rigid symmetry. The observed distances break symmetry and equidistance, rejecting the possibility of forming a regular tetrahedron’s face.", "### Conclusion: These Points Are Not Regular Tetrahedron Vertices", "In summary, three points with $AB = BC = 3\sqrt{3}$ and $AC = 6\sqrt{3}$ violate the essential requirement of equal edge lengths in equilateral triangular faces. As a result, they cannot be vertices of a regular tetrahedron, whether embedded in 3D space or constrained to integer-coordinate planes. Their geometry defies the fundamental symmetry and uniformity expected in regular tetrahedral structures.", "This insight underscores the importance of validating both spatial relationships and geometric consistency when analyzing potential configurations in geometry and 3D modeling — particularly when integer coordinates or symmetry are assumed.", "---", "Keywords: regular tetrahedron face geometry, equilateral triangle distances, $AB = BC = 3\sqrt{3}$, $AC = 6\sqrt{3}$, non-equidistant points, integer coordinates constraint, 3D geometry, planar triangle impossibility, tetrahedron vertex validity", "Meta description: Explains why a triangle with sides $3\sqrt{3}, 3\sqrt{3}, 6\sqrt{3}$ cannot form a face of a regular tetrahedron — due to violation of equidistance and triangle symmetry. Includes geometric proof and implications for integer-coordinate vertices."]
