Proposition 7
Theorem
If a line segment AB is divided into any two parts at point C, then 2 ￜAB ￜ⋅ￜCB ￜ + ￜAC ￜ2 = ￜAB ￜ2 + ￜCB ￜ2 .
Commentary
1. Let AB be the given line segment, and let C be any point on AB .
2. OnAB , construct a square ◇ABDE .
3. Find a point G onED such that CG is parallel to the side AE of the square, and intersects the diagonal BE at a point F.
4. Through F, draw a lineHK that is parallel to AB .
5. The rectangle◇ABKH has been constructed with the area equal to that of rectangle ◇CBDG .
6. The sum of the areas of rectangles◇ABKH (ￜAB ￜ⋅ￜCB ￜ ), ◇CBDG (ￜAB ￜ⋅ￜCB ￜ ), and square ◇HFGE (ￜAC ￜ2 ) is equal to the sum of the areas of the two squares: ◇ABDE (ￜAB ￜ2 ) and ◇CBKF (ￜCB ￜ2 ).
7. This geometric relationship can be expressed algebraically as follows: ifx = y + z , then 2 x z + y2 = x2 + z2 (where ￜAB ￜ = x , ￜAC ￜ = y and ￜCB ￜ = z ). This algebraic relationship is a polynomial identity.
2. On
3. Find a point G on
4. Through F, draw a line
5. The rectangle
6. The sum of the areas of rectangles
7. This geometric relationship can be expressed algebraically as follows: if
Original statement
ἐὰν ϵὐθϵῖα γραμμὴ τμηθῇ, ὡς ἔτυχϵν, τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀϕ᾽ ἑνὸς τῶν τμημάτων τὰ συναμϕότϵρα τϵτράγωνα ἴσα ἐστὶ τῷ τϵ δὶς ὑπὸ τῆς ὅλης καὶ τοῦ ϵἰρημένου τμήματος πϵριϵχομένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τοῦ λοιποῦ τμήματος τϵτραγώνῳ.
English translation
If a straight line is cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said segment and the square on the remaining segment.
