Geometric Construction of Squares on Grid Paper by orchiming

This GeoGebra applet demonstrates techniques for precisely constructing squares on grid paper or geoboards. It leverages mathematical tools including the Pythagorean theorem, vectors (unit vectors and orthogonal vectors), rotation transformations, and the greatest common divisor (GCD) to systematically determine vertex coordinates of squares whose sides may not align with the grid axes. The construction features sequence generation, polygon drawing, right-angle and equal-side markers, and parallel-side indicators as essential geometric construction elements. Through interactive manipulation, learners can explore how斜置正方形（tilted squares）emerge naturally on a grid by selecting appropriate vector pairs that satisfy both length equality and orthogonality conditions. The applet visually connects abstract algebraic concepts—such as vector dot products and coordinate geometry—with concrete geometric visualization, reinforcing the relationship between integer coordinate points and the side lengths derived via the Pythagorean theorem. Students investigate how the GCD of coordinate differences governs whether a segment passes through additional lattice points, deepening their understanding of number-theoretic constraints in geometric constructions. This applet is well-suited for middle school to high school geometry instruction, supporting inquiry-based learning activities where students predict, verify, and generalize patterns related to squares on grids. By combining dynamic visual feedback with rigorous mathematical reasoning, the tool helps learners develop spatial intuition, strengthen proof-writing skills, and appreciate the interconnectedness of algebra and geometry.