Geometric Construction of Squares and Xian Tu

This applet demonstrates the geometric construction of squares and the Xian Tu, a classical Chinese diagram named after ancient mathematician Zhao Shuang. Through interactive steps involving polygon construction, line segments, and intersection points, the model dynamically illustrates the area relationships and geometric transformations between right triangles and surrounding squares. The Xian Tu is a powerful visual proof of the Pythagorean theorem, showing how four congruent right triangles can be arranged within a larger square to reveal that the sum of the areas of the two smaller squares equals the area of the largest square. Students can manipulate the dimensions of the right triangle and observe in real time how the configuration transforms while preserving the fundamental area equality. The applet also explores properties of inscribed figures within squares, revealing symmetries and proportional relationships that deepen understanding of Euclidean geometry. This interactive demonstration serves as an effective teaching tool for plane geometry classrooms, allowing learners to move beyond static textbook diagrams and engage with the proof through hands-on exploration. By visually connecting algebraic concepts to geometric intuition, the applet helps students internalize one of mathematics most celebrated theorems and appreciate the cultural heritage embedded in its historical origins.