But the question says "the node opposite to $O$", i.e., opposite the origin. That wording is misleading — $O$ is at the origin. But in the hexagon, the node farthest from $O$ is at $(-1, 0)$, which is at distance 1 — same as all others. But in a **regular hexagon**, all vertices are equidistant from center, so **no node is farther than others**.

["The Truth About the Node Opposite to $O$ in a Regular Hexagon: Why All Vertices Are Equidistant from the Center", "When analyzing geometric shapes like regular hexagons, precise terminology matters—especially when it comes to distance, position, and symmetry. A common point of confusion arises from phrases like “the node opposite to $O$”, where $O$ represents the center (origin) of a hexagon. This phrasing can misleadingly suggest that, unlike other vertices, a specific “opposite” node stands apart in distance from the origin. But in reality, in a regular hexagon, every vertex lies exactly the same distance from $O$. Let’s clarify this common misconception and reveal the true symmetry behind the witchcraft of geometry.", "---", "### What Is a Regular Hexagon—and Why All “Nodes” Are Equal", "A regular hexagon is a polygon with six identical sides and six equal angles. Mathematically, its six vertices lie exactly on a circle centered at $O$, equidistant from the origin. In coordinate geometry, a standard regular hexagon centered at the origin often places vertices at angles $0^\circ, 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ$ on the unit circle. The node at $(-1, 0)$—commonly labeled as the “opposite” node to the origin’s neighbors—has the same radial distance (radius = 1) as every other vertex.", "Thus, there is no node farther from $O$ than the others—they are all equidistant. The symmetry ensures each “node” is formally the same, rotating by $60^\circ$ increments.", "---", "### Why the Misleading Use of “Opposite”?", "The term “opposite” typically refers to the farthest point along a diameter. In a circle or regular polygon centered at $O$, opposite means directly across the center—at $180^\circ$, for instance, the node at $(1, 0)$ lies opposite $(-1, 0)$. Since all vertices are equidistant from $O$, there is no inherent “opposite” node with greater distance. Calling one node “opposite” doesn’t imply increasing radial distance—it reflects angular positioning, not magnitude.", "Understanding this avoids confusion and emphasizes the true beauty: rotational symmetry. Rotating the hexagon in any direction preserves all vertex distances from $O$. No node outshines the others; each plays a role in a perfectly balanced figure.", "---", "### Geometric Insight: All Vertices Are Equidistant", "If the hexagon is centered at the origin $(0,0)$ and defined with a radius $r$, every vertex $(x, y)$ satisfies:", "$$\nx^2 + y^2 = r^2\n$$", "For a unit regular hexagon, $r = 1$. Hence, each vertex lies on the unit circle: it’s distance $\sqrt{x^2 + y^2} = 1$ from $O$. Therefore, no vertex is farther from $O$ than the others.", "---", "### Conclusion: Embracing True Symmetry", "The idea of a “node opposite to $O$” that is farther away is a misconception rooted in angular reasoning rather than radial distance. In a regular hexagon, symmetry makes all vertices equal in distance from the center. Recognizing this symmetry deepens our appreciation of geometric harmony—where balance, repetition, and rotational invariance reign, and no node stands taller than any other.", "So remember: in the regular hexagon, every node is equally “far” from $O$, and the true magic lies not in opposition of distance, but in the perfect uniformity that defines regular polygons.", "---", "Keywords: regular hexagon, farthest node from origin, distance from center hexagon, symmetric vertices, origin geometry, radial symmetry, unit circle hexagon, angular positioning vs radial distance, equidistant nodes, hexagon symmetry, $O$ center geometry", "Meta Description: Discover why in a regular hexagon, all nodes are equidistant from the center $O$—no node is truly farther, thanks to perfect rotational symmetry and equal radius from origin. Learn geometric truth behind misleading “opposite” node claims."]