Dowód Geometryczny Euklidesa („Wiatrak”)
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.
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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).
Kontekst Historyczny: Księga I, Twierdzenie 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”.
Euklides vs Bhaskara vs Algebra
| 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 |