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

For a right triangle $A B C$ with the right angle at $C$ , legs $a = B C$ and $b = C A$ , and hypotenuse $c = A B$ :

a^{2} + b^{2} = c^{2}

Strategy: build a square on each side, then drop the altitude from $C$ to the hypotenuse, extending it to split the blue square on $c$ into two rectangles. Congruent triangles show the red square ( $b^{2}$ ) equals one rectangle and the green square ( $a^{2}$ ) equals the other, so together they fill the blue square.

Step 1: Right Triangle $A B C$

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

Step 2: Construct Squares on Each Side

We construct squares on each side of the triangle: a green square on leg $a$ (side $B C$ ), a red square on leg $b$ (side $C A$ ), and a blue square on the hypotenuse $c$ (side $A B$ ). 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 $A B$ . Call the foot of this altitude point $F$ . Then extend this altitude through the square on $A B$ to point $L$ .

Step 5: The Two Rectangles

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

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

Draw segments $B E$ and $C J$ . We will show that $△ A E B ≅ △ A C J$ by SAS:

$A B = A J = c$ (both sides of the blue square)
$A E = C A = b$ (both sides of the red square)
$∠ B A E = ∠ C A J$ (both are $90° + ∠ B A C$ )

Since $△ A E B$ has half the area of the red square, and $△ A C J$ has half the area of rectangle $A J L F$ , and the triangles are congruent: Red square ( $b^{2}$ ) = Rectangle $A J L F$ .

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

Similarly, draw segments $A G$ and $C K$ . We can show that $△ B G A ≅ △ B C K$ by SAS:

$B G = B C = a$ (both sides of the green square)
$B A = B K = c$ (both sides of the blue square)
$∠ GB A = ∠ C B K$ (both are $90° + ∠ A B C$ )

Since $△ B G A$ has half the area of the green square, and $△ B C K$ has half the area of rectangle $F L K B$ : Green square ( $a^{2}$ ) = Rectangle $F L K B$ .

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

The two rectangles together equal the blue square on the hypotenuse. But rectangle $A J L F$ = red square ( $b^{2}$ ) and rectangle $F L K B$ = green square ( $a^{2}$ ). Therefore: $a^{2} + b^{2} = c^{2}$ . This completes Euclid's proof of the Pythagorean theorem!

We have shown:

Red square ( $b^{2}$ ) = Rectangle $A J L F$
Green square ( $a^{2}$ ) = Rectangle $F L K B$
Rectangle $A J L F$ + Rectangle $F L K B$ = Blue square ( $c^{2}$ )

Therefore: $a^{2} + b^{2} = c^{2}$ . ∎

Notes

This proof is Euclid's Elements, Book I, Proposition 47, written around 300 BCE. With all three squares drawn the figure resembles a windmill, giving it the nicknames "Euclid's Windmill" and "The Bride's Chair."

Historical Note

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

Why This Proof Matters

Euclid's proof uses only geometry — no algebra or arithmetic — and shows a direct area relationship: you can literally see how the pieces fit together, with congruent triangles establishing that equal triangles imply equal areas. It has survived 2,300+ years as one of the oldest complete proofs we have.