Proof 1 of Pythagorean Theorem: Euclid's Windmill

Euclid's famous proof (Elements Book I, Proposition 47) showing that the area of the square on the hypotenuse equals the sum of the areas of squares on the legs. The construction resembles a windmill, using congruent triangles to match areas.

Prerequisites: pythagorean-theorem, triangle-congruence · Difficulty: intermediate

This is a famous proof from Euclid's Elements (Book I, Proposition 47), written around 300 BCE. The construction resembles a windmill when all the squares are drawn, giving it the nickname "Euclid's Windmill" or "The Bride's Chair."

The Setup

Start with a right triangle $ABC$ where:

• $C$ is the vertex with the right angle (90°)
• Side $a = BC$ is one leg
• Side $b = CA$ is the other leg
• Side $c = AB$ is the hypotenuse (opposite the right angle)

Construct squares on each side: a green square on side $a$, a red square on side $b$, and a blue square on the hypotenuse $c$.

The Key Construction

From the right angle vertex $C$, drop a perpendicular to the hypotenuse $AB$, meeting it at point $F$. Extend this line through the blue square to point $L$. This line divides the blue square into two rectangles: Rectangle $AJLF$ (adjacent to $A$) and Rectangle $FLKB$ (adjacent to $B$).

Proof Strategy

Euclid proves two key facts:

1. The red square ($b^2$) has the same area as rectangle $AJLF$
2. The green square ($a^2$) has the same area as rectangle $FLKB$

Since the two rectangles together make up the entire blue square, this proves $a^2 + b^2 = c^2$.

The Proof

Step 1: Right Triangle $ABC$

We begin with a right triangle $ABC$ where the right angle is at vertex $C$. The two legs are $a = BC$ and $b = CA$, and the hypotenuse is $c = AB$.

Step 2: Construct Squares on Each Side

We construct squares on each side of the triangle: a green square on leg $a$ (side $BC$), a red square on leg $b$ (side $CA$), and a blue square on the hypotenuse $c$ (side $AB$). This construction is sometimes called "Euclid's Windmill" because of its appearance.

Step 3: Pythagorean Theorem

Proof strategy (Euclid's Windmill). Drop the altitude from $C$ to the hypotenuse, splitting the blue square into two rectangles. Use congruent triangles to show each rectangle equals one of the smaller squares — so $a^2 + b^2 = c^2$.

Step 4: Draw the Altitude from $C$

The key construction: draw the altitude from the right angle vertex $C$ perpendicular to the hypotenuse $AB$. Call the foot of this altitude point $F$. Then extend this altitude through the square on $AB$ to point $L$.

Step 5: The Two Rectangles

The line $FL$ divides the square on the hypotenuse into two rectangles: rectangle $AJLF$ (adjacent to vertex $A$) and rectangle $FLKB$ (adjacent to vertex $B$). We will prove that the red square equals rectangle $AJLF$, and the green square equals rectangle $FLKB$.

Step 6: Part 1: Red Square = Rectangle (Congruent Triangles)

Draw segments $BE$ and $CJ$. We will show that $\triangle AEB \cong \triangle ACJ$ by SAS:

• $AB = AJ = c$ (both sides of the blue square)
• $AE = CA = b$ (both sides of the red square)
• $\angle BAE = \angle CAJ$ (both are $90° + \angle BAC$)

Since $\triangle AEB$ has half the area of the red square, and $\triangle ACJ$ has half the area of rectangle $AJLF$, and the triangles are congruent: Red square ($b^2$) = Rectangle $AJLF$.

Step 7: Part 2: Green Square = Rectangle (Congruent Triangles)

Similarly, draw segments $AG$ and $CK$. We can show that $\triangle BGA \cong \triangle BCK$ by SAS:

• $BG = BC = a$ (both sides of the green square)
• $BA = BK = c$ (both sides of the blue square)
• $\angle GBA = \angle CBK$ (both are $90° + \angle ABC$)

Since $\triangle BGA$ has half the area of the green square, and $\triangle BCK$ has half the area of rectangle $FLKB$: Green square ($a^2$) = Rectangle $FLKB$.

Step 8: The Conclusion: $a^2 + b^2 = c^2$

The two rectangles together equal the blue square on the hypotenuse. But rectangle $AJLF$ = red square ($b^2$) and rectangle $FLKB$ = green square ($a^2$). Therefore: $a^2 + b^2 = c^2$. This completes Euclid's proof of the Pythagorean theorem!

Conclusion

We have shown:

• Red square ($b^2$) = Rectangle $AJLF$
• Green square ($a^2$) = Rectangle $FLKB$
• Rectangle $AJLF$ + Rectangle $FLKB$ = Blue square ($c^2$)

Therefore: $a^2 + b^2 = c^2$.

Why This Proof Matters

Euclid's proof is remarkable because it:

1. Uses only geometry — no algebra or arithmetic needed
2. Shows a direct area relationship — you can literally see how the pieces fit together
3. Demonstrates the power of congruent triangles — equal triangles imply equal areas
4. Has survived 2,300+ years — it's one of the oldest complete proofs we have

Historical Note

While the theorem is named after Pythagoras (c. 570–495 BCE), Euclid's proof is the first complete written proof that survives to modern times. The Pythagorean theorem was known even earlier in Babylon, Egypt, India, and China, but Euclid's rigorous proof set the standard for mathematical demonstration.