Solution: The given points lie on the base face of the cube in the $xy$-plane. The fourth vertex of the square face must complete the rectangle. The missing vertex is $(1,1,0)$, as it forms equal sides and right angles with the given points.

["The Missing Vertex on the Square Face of a Cube in the $xy$-Plane", "When visualizing a cube embedded in the $xy$-plane, one essential geometric fact emerges: the square faces of the cube align precisely with flat, level bases. In this scenario, we are told that four of the vertices lie on the base face within the $xy$-plane, specifically at the points $(0,0,0)$, $(0,1,0)$, $(1,1,0)$, and an unknown fourth point. We aim to determine this missing vertex to complete the square face—crucial for understanding the cube’s full structure.", "The cube’s base lies flat on the $xy$-plane, meaning all its vertices have a $z$-coordinate of 0. Given the known points $(0,0,0)$, $(0,1,0)$, and $(1,1,0)$, we notice that these three points define a perfect right angle at $(0,1,0)$ and form adjacent sides of the square. The length between $(0,0,0)$ and $(0,1,0)$ is 1 unit, and the segment from $(0,1,0)$ to $(1,1,0)$ is also 1 unit—confirming equal adjacent edges.", "To complete the square, we need a fourth point that maintains equal side lengths and right angles with the adjacent vertices. Since moving from $(1,1,0)$ along a direction parallel to the $x$-axis for one unit brings us to $(1,0,0)$, we examine the expected fourth corner.", "However, the problem specifically identifies the missing vertex as $(1,1,0)$. Wait—this appears inconsistent. Let’s clarify: given $(0,0,0)$, $(0,1,0)$, and $(1,1,0)$, the missing vertex completing the square base should be $(1,0,0)$, not $(1,1,0)$—that point is already among the listed known vertices.", "Instead, the key insight lies in recognizing the correct fourth vertex required to close the square sequence: starting at $(0,0,0) \rightarrow (0,1,0) \rightarrow (1,1,0)$, the next logical corner to close the rectangle faces $(1,0,0)$. Yet, the problem’s phrasing suggests a focus on completing the square using orthogonal projection and equal side validation.", "But upon re-evaluating, if the three given points are $(0,0,0)$, $(0,1,0)$, and $(1,1,0)$, their configuration confirms a unit square rotated into the $xy$-plane with all sides length 1 and right angles at each joint. The fourth and missing vertex forming the opposite corner of the base square is indeed $(1,0,0)$—completing equal side lengths and right angles throughout.", "However, the prompt states the missing vertex is $(1,1,0)$, which contradicts geometry unless interpreted differently—perhaps as a mislabeled vertex. Alternatively, if the three known points are $(0,0,0)$, $(0,1,0)$, and $(1,1,0)$, then the correct missing fourth corner completing the base square is $(1,0,0)$, ensuring:", "- Vector lengths: $|(0,1,0) - (0,0,0)| = 1$, $|(1,1,0) - (0,1,0)| = 1$, $|(1,0,0) - (1,1,0)| = 1$, and $|(0,0,0) - (1,0,0)| = 1$\n- Adjacent sides meet at $90^\circ$ angles\n- Opposite sides are parallel, forming a complete rectangle", "Thus, the square face lies entirely in the $xy$-plane with vertices at $(0,0,0)$, $(0,1,0)$, $(1,1,0)$, and $(1,0,0)$—the fourth vertex completing the rectangle/face is $(1,0,0)$. The claim that the missing vertex is $(1,1,0)$ seems erroneous unless interpreted as a point overlapping a known vertex, reinforcing that $(1,1,0)$ cannot be the fourth corner in this set.", "Therefore, the correct solution is: the missing fourth vertex is $(1,0,0)$, which completes the square base face in the $xy$-plane with equal side lengths and right angles. This aligns with standard geometric principles for cube faces lying on coordinate planes.", "In summary, for a cube’s base face in the $xy$-plane, with three vertices at $(0,0,0)$, $(0,1,0)$, and $(1,1,0)$, the missing vertex forming a complete square (and thus a base face) is $(1,0,0)$. Recognizing correct spatial relationships ensures accurate modeling of cube geometry, vital in 3D design, engineering, and mathematics."]