and between $(1,1,0)$ and $(1,0,1)$: $\sqrt{(0)^2 + (-1)^2 + (1)^2} = \sqrt{2}$, etc. All edges $\sqrt{2}$. Thus, the fourth vertex is $(0,1,1)$.

["Understanding the 3D Geometry Between Points: From $(1,1,0)$ and $(1,0,1)$ to the Complete Tetrahedron with Edge Length $\sqrt{2}$", "Web 3D geometry often invites exploration of spatial relationships, especially when analyzing vertices of polyhedra like tetrahedrons. One intriguing example involves comparing three coordinate points — $(1,1,0)$, $(1,0,1)$, and $(1,0,1)$ — and determining the final fourth vertex that completes a regular tetrahedron with equal edge lengths of $\sqrt{2}$. By breaking down the distances and geometry step by step, we uncover how the missing vertex $(0,1,1)$ satisfies all edge conditions.", "In a regular tetrahedron, all six connecting edges are congruent. Here, we are given two initial points: $A = (1,1,0)$ and $B = (1,0,1)$. The distance between these points is calculated using the Euclidean formula:", "$$\nAB = \sqrt{(1 - 1)^2 + (1 - 0)^2 + (0 - 1)^2} = \sqrt{0 + 1 + 1} = \sqrt{2}\n$$", "Thus, the edge length is confirmed as $\sqrt{2}$. The goal is to find point $C = (x,y,z)$ such that all edges from $C$ to $A$ and $B$, as well as $A$ to $B$, measure exactly $\sqrt{2}$. This demands $|CA| = |CB| = \sqrt{2}$ and symmetry consistent with a regular tetrahedron.", "To determine the coordinates of $C = (0,1,1)$, we verify each required edge length:", "Distance $CA$:\n$$\nCA = \sqrt{(0 - 1)^2 + (1 - 1)^2 + (1 - 0)^2} = \sqrt{(-1)^2 + 0 + 1^2} = \sqrt{1 + 0 + 1} = \sqrt{2}\n$$", "Distance $CB$:\n$$\nCB = \sqrt{(0 - 1)^2 + (1 - 0)^2 + (1 - 1)^2} = \sqrt{1 + 1 + 0} = \sqrt{2}\n$$", "Distance $AB$: Already confirmed as $\sqrt{2}$.", "Moreover, symmetry and geometric principles confirm that $(0,1,1)$ forms the fourth vertex of a regular tetrahedron with $(1,1,0)$ and $(1,0,1)$. The full set of edge lengths remains consistent, with each pair of vertices separated by exactly $\sqrt{2}$. This illustrates a classic example of using coordinate geometry to satisfy edge constraints in 3D space — a key skill for applications in computer graphics, architecture, and machine learning.", "In conclusion, the tetrahedron with vertices $(1,1,0)$, $(1,0,1)$, and $(0,1,1)$ — connected by all $\sqrt{2}$ edges — exemplifies symmetry and precision in 3D geometry. Understanding such configurations empowers deeper spatial reasoning and supports advanced spatial analysis in both theoretical and applied domains.", "---", "Key Takeaways:", "- The distance formula in 3D space allows exact edge length calculation.\n- Points forming edges of equal length $\sqrt{2}$ can compose regular polyhedra.\n- The vertex $(0,1,1)$ satisfies all required $\sqrt{2}$ edge lengths with $(1,1,0)$ and $(1,0,1)$.\n- This exemplifies how coordinate geometry enables verification of symmetric spatial arrangements.\n- Strong geometric reasoning is essential for computer-aided design, robotics, and visualization.", "By mastering such spatial relationships, students and professionals alike enhance their ability to model and analyze real-world 3D systems efficiently."]