Pembuktian Geometri Euclid ('Kincir Angin')
Written around 300 BCE in Alexandria, Euclid’s Elements Book I, Proposition 47 proves a² + b² = c² purely by chopping the hypotenuse square into two rectangles matched identically to the leg squares—no algebraic numbers required. Looking to calculate numerical side lengths instead? Launch our Pythagoras equation solver.
1. Bangun Persegi di Ketiga Sisi Segitiga
Given right-angled triangle △ABC (∠ACB = 90°), construct exterior squares on all three sides: BC² = CBHK (area a²), AC² = ACGF (area b²), and hypotenuse AB² = ABED (area c²).
Konteks Sejarah: Euclid Buku I, Proposisi 47
Unlike algebraic proofs that manipulate (a + b)² and assume irrational lengths can always be multiplied as decimals, Greek geometry faced the crisis of incommensurable lengths like √2. Euclid avoided proportion ratios entirely until Book V—so Book I, Proposition 47 proves the theorem using only straightedge-and-compass axioms: parallel bases, half-area triangles, and Side-Angle-Side (SAS) congruence.
Because the three outward squares around triangle △ABC together with the internal dividing rays resemble a rotating turbine or bridal headgear, medieval scholars nicknamed Proposition I.47 “The Windmill” or “The Bride’s Chair”.
Euclid vs Bhaskara vs Bukti Aljabar
| Proof Family | Core Geometric Tool | Uses Algebra / Numbers? | Best Intuition |
|---|---|---|---|
| Euclid’s Windmill (I.47) | SAS triangle congruence + area shear between parallels | No (Pure Synthetic) | Links each leg square directly to one rectangle slice of c² |
| Bhaskara’s Rearrangement | Conserving leftover space inside an (a + b)² tray | Optional / Visual | Instant “Behold!” visual conservation of 4 triangle tiles |
| Algebraic / Similarity Proofs | (a + b)² expansion or right-triangle similarity ratios | Yes (Quadratic / Ratios) | Fastest 2-line symbolic proof in coordinate & vector math |