Proposition 2
Theorem
If a line AB is divided into two parts at point C, then ￜAB ￜ2 = ￜAB ￜ⋅ￜAC ￜ + ￜAB ￜ⋅ￜCB ￜ .
Commentary
1. Let AB be the given line segment, and let C be any point on AB .
2. Construct a square◇ABDF on AB .
3. Pick a point E onDF and draw EC perpendicular to DF so that EC has the same length as AB and EC is a shared side of ◇ACEF and ◇CBDE .
4. The area of the square◇ABDF (ￜAB ￜ2 ) is the sum of the areas of the two rectangles ◇ACEF (ￜAB ￜ⋅ￜAC ￜ ) and ◇CBDE (ￜAB ￜ⋅ￜCB ￜ ).
5. This geometric relationship can be expressed algebraically as follows: ifx = a + b , then x2 = x a + x b (where ￜAC ￜ = a , ￜCB ￜ = b and ￜAB ￜ = x ). This algebraic relationship is a special case of the distributive law.
2. Construct a square
3. Pick a point E on
4. The area of the square
5. This geometric relationship can be expressed algebraically as follows: if
Original statement
ἐὰν ϵὐθϵῖα γραμμὴ τμηθῇ, ὡς ἔτυχϵν, τὸ ὑπὸ τῆς ὅλης καὶ ἑκατέρου τῶν τμημάτων πϵριϵχόμϵνον ὀρθογώνιον ἴσον ἐστὶ τῷ ἀπὸ τῆς ὅλης τϵτραγώνῳ.
English translation
If a straight line is cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole.
