Proposition 8
Theorem
If a line segment AB is divided into any two parts at point C, then 4 ￜAB ￜ⋅ￜCB ￜ + ￜAC ￜ2 = ￜAB ￜ + ￜCB ￜ2 .
Commentary
1. Let AB be the given line segment, and let C be any point on AB .
2. ExtendAB to D so that B is the midpoint of line CD .
3. Construct square◇ADFE on AD , and find a point H on EF such that CH is parallel to AE and intersects the diagonal DE at a point I.
4. Find a point L onEF such that BL is parallel to AE and intersects the diagonal DE at a point K.
5. Through I, draw a line parallel toAD and intersecting AE , BL and DF at points P, J and O, respectively.
6. Through K, draw a line parallel toAD and intersecting AE , CH and DF at points M, G and N, respectively.
7. These constructed lines divide the square◇ADFE into nine polygons as follows: four rectangles ◇ACGM , ◇MGIP , ◇IJLH , and ◇JOFL with the same area; four squares ◇CBKG , ◇GKJI , ◇BDNK , and ◇KNOJ with the same area; and a square ◇PIHE .
8. The sum of four times the area of rectangle◇ABKM (4 ￜAB ￜ⋅ￜCB ￜ ) and the area of square ◇PIHE (ￜAC ￜ2 ) is equal to the area of square ◇ADFE (ￜAB ￜ + ￜBC ￜ2 ).
9. This geometric relationship is similar to the previous proposition, Book 2 Proposition 7, and can also be expressed algebraically as follows: ifx = y + z , then 4 x y + z2 = x + y2 (where ￜAB ￜ = x , ￜBC ￜ = y and ￜAC ￜ = z ). This algebraic relationship is a polynomial identity.
2. Extend
3. Construct square
4. Find a point L on
5. Through I, draw a line parallel to
6. Through K, draw a line parallel to
7. These constructed lines divide the square
8. The sum of four times the area of rectangle
9. This geometric relationship is similar to the previous proposition, Book 2 Proposition 7, and can also be expressed algebraically as follows: if
Original statement
ἐὰν ϵὐθϵῖα γραμμὴ τμηθῇ, ὡς ἔτυχϵν, τὸ τϵτράκις ὑπὸ τῆς ὅλης καὶ ἑνὸς τῶν τμημάτων πϵριϵχόμϵνον ὀρθογώνιον μϵτὰ τοῦ ἀπὸ τοῦ λοιποῦ τμήματος τϵτραγώνου ἴσον ἐστὶ τῷ ἀπό τϵ τῆς ὅλης καὶ τοῦ ϵἰρημένου τμήματος ὡς ἀπὸ μιᾶς ἀναγραϕέντι τϵτραγώνῳ.
English translation
If a straight line is cut at random, four times the rectangle contained by the whole and one of the segments together with the square on the remaining segment is equal to the square described on the whole and the aforesaid segment as on one straight line.
