Proposition 47
Theorem
Alternate name(s): Pythagorean theorem, bride's chair, windmill.
In a right triangle (△ABC ) the area of the square on the hypotenuse (AB ) is equal to the sum of the areas of the squares on the other two sides (AC , BC ).
Commentary
1. Let △ABC be a given right triangle with right angle at C.
2. On each side of△ABC , let a square be constructed.
3. The area of the square on the hypotenuse (AB ) is the sum of the areas of the two squares on the other two sides (AC and BC ).
4. This geometric relationship between the areas of the constructed squares is often expressed algebraically asa2 + b2 = c2 , meaning ￜACￜ2 + ￜBCￜ2 = ￜABￜ2 .
5. This proposition is generally known as the Pythagorean theorem and Book 1 Proposition 48 is the converse.
2. On each side of
3. The area of the square on the hypotenuse (
4. This geometric relationship between the areas of the constructed squares is often expressed algebraically as
5. This proposition is generally known as the Pythagorean theorem and Book 1 Proposition 48 is the converse.
Original statement
ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτϵινούσης πλϵυρᾶς τϵτράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν πϵριϵχουσῶν πλϵυρῶν τϵτραγώνοις.
English translation
In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
