Solution: Since the base triangle lies in the $ xy $-plane and is equilateral (verified by equal side lengths of 4), the fourth vertex lies directly above the centroid of the base triangle. The centroid $ G $ is:

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**Understanding the Geometry Behind Equilateral Triangles: The Role of the Centroid (and the Hidden Fourth Vertex)** Ever wondered how precise shapes form in nature, design, or digital models? One fascinating pattern emerges when a triangle lies flat in the $ xy $-plane with equal sides—specifically, an equilateral triangle. Recent curiosity online reveals how this balanced form naturally guides the placement of a fourth point: directly above the centroid when a perpendicular line is drawn. This solution isn’t just geometrically elegant—it’s a quiet building block in fields from architecture to data visualization. **Growing Interest in Spatial Accuracy** With rising interest in spatial design, architectural modeling, and precise measurement tools, understanding triangle geometry has never felt more relevant. Social trends highlight how balanced, mathematically sound structures influence user trust and digital aesthetics. Readers exploring design principles, STEM learning, or interactive visualizations increasingly seek foundational geometric solutions—like why the fourth vertex in an equilateral triangle rises straight up from its centroid.