Proposition 3
Theorem
If a line AB is divided into two parts at point C, then ￜAB ￜ⋅ￜCB ￜ = ￜCB ￜ2 + ￜAC ￜ⋅ￜCB ￜ .
Commentary
1. Let AB be the given line segment, and let C be any point on AB .
2. OnBC , construct a square ◇CBED .
3. ExtendED to F to construct rectangles ◇ABEF and ◇ACDF .
4. The area of the rectangle◇ABEF (ￜAB ￜ⋅ￜCB ￜ ) is the sum of the areas of the two rectangles ◇CBED (ￜCB ￜ2 ) and ◇ACDF (ￜAC ￜ⋅ￜCB ￜ ).
5. This geometric relationship can be expressed algebraically as follows: ifx = y + z , then x y = y2 + y z (where ￜAB ￜ = x , ￜCB ￜ = y and ￜAC ￜ = z ). This algebraic relationship is a special case of the distributive law.
2. On
3. Extend
4. The area of the rectangle
5. This geometric relationship can be expressed algebraically as follows: if
Original statement
ἐὰν ϵὐθϵῖα γραμμὴ τμηθῇ, ὡς ἔτυχϵν, τὸ ὑπὸ τῆς ὅλης καὶ ἑνὸς τῶν τμημάτων πϵριϵχόμϵνον ὀρθογώνιον ἴσον ἐστὶ τῷ τϵ ὑπὸ τῶν τμημάτων πϵριϵχομένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τοῦ προϵιρημένου τμήματος τϵτραγώνῳ.
English translation
If a straight line is cut at random, the rectangle contained by the whole and one of the segments is equal to the rectangle contained by the segments and the square on the aforesaid segment.
