Zhao Shuang was a brilliant Chinese mathematician from the third century CE, living from 220 to 280. He created an elegant geometric proof of the Pythagorean theorem using his famous Chord Diagram. This visual demonstration predates many Western proofs of this fundamental mathematical relationship, showcasing the rich history of mathematical discovery across cultures.
The Zhao Shuang chord diagram is constructed by arranging four identical right triangles around a central square. We start with a right triangle having sides a and b, and hypotenuse c. Four copies of this triangle are then arranged around a central square with side length c, forming a larger square with side length a plus b. This elegant geometric arrangement creates the foundation for proving the Pythagorean theorem.
The proof works by calculating the total area in two different ways. The outer square has area a plus b squared, while it also equals the sum of four triangles, each with area one half a b, plus the inner square with area c squared. This gives us the equation: a plus b squared equals four times one half a b plus c squared. Expanding the left side gives a squared plus two a b plus b squared equals two a b plus c squared. Subtracting two a b from both sides yields the Pythagorean theorem: a squared plus b squared equals c squared.
Now let's see how the algebraic steps correspond to the geometric parts. The outer square represents a plus b squared. The four triangles each contribute one half a b, totaling two a b. The inner square contributes c squared. When we expand a plus b squared, we get a squared plus two a b plus b squared. Notice how the two a b terms appear on both sides of the equation and cancel out, leaving us with the beautiful Pythagorean theorem: a squared plus b squared equals c squared.
Let's verify the theorem with the famous 3-4-5 right triangle. Each triangle has area one half times 3 times 4, which equals 6. The inner square has area 5 squared, which equals 25. The outer square has area 7 squared, which equals 49. We can verify: four times 6 plus 25 equals 24 plus 25, which equals 49. Most importantly, we confirm that 3 squared plus 4 squared equals 9 plus 16, which equals 25, which equals 5 squared. This numerical example perfectly demonstrates the Pythagorean theorem.