Proposition 4
Theorem
If a line AB is divided into two parts at point C, then ￜABￜ2 = ￜACￜ2 + ￜCBￜ2 + 2 ￜACￜ⋅ￜCBￜ .
Commentary
1. Let AB be the given line segment, and let C be any point on AB .
2. OnAB , construct a square ◇ABDE .
3. OnDE , find a point F such that CF is parallel to the side BD of square ◇ABDE and intersects the diagonal BE at a point G.
4. Through G draw a lineHI parallel to AB .
5. This construction divides square◇ABDE into four parts: two squares ◇HGFE and ◇CBIG , and two rectangles ◇ACGH and ◇GIDF with the same area.
6. The area of the square◇ABDE (ￜABￜ2 ) is the sum of the four parts: ◇HGFE (ￜACￜ2 ), ◇CBIG (ￜCBￜ2 ), ◇ACGH (ￜACￜ⋅ￜCBￜ ) and ◇GIDF (ￜACￜ⋅ￜCBￜ ).
7. This geometric relationship can be expressed algebraically as follows: ifx = y + z then x2 = y2 + z2 + 2 y z (where ￜABￜ = x , ￜACￜ = y and ￜCBￜ = z ). This algebraic relationship is the special case of the binomial theorem for exponent 2.
2. On
3. On
4. Through G draw a line
5. This construction divides square
6. The area of the square
7. This geometric relationship can be expressed algebraically as follows: if
Original statement
ἐὰν ϵὐθϵῖα γραμμὴ τμηθῇ, ὡς ἔτυχϵν, τὸ ἀπὸ τῆς ὅλης τϵτράγωνον ἴσον ἐστὶ τοῖς τϵ ἀπὸ τῶν τμημάτων τϵτραγώνοις καὶ τῷ δὶς ὑπὸ τῶν τμημάτων πϵριϵχομένῳ ὀρθογωνίῳ.
English translation
If a straight line is cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
