Proposition 6
Theorem
If a line AB is bisected at point C, and divided externally at any point D, then ￜAD ￜ⋅ￜBD ￜ + ￜCB ￜ2 = ￜCD ￜ2 .
Commentary
1. Let AB be the given line segment. Let C be the midpoint of AB , and let D be any point on the extension of AB .
2. Construct a square◇CDFE on CD , and find a point G on EF such that BG is parallel to DF and intersects the diagonal DE at a point H.
3. Through H, draw a line parallel toAD and intersecting CE and DF at L and M, respectively.
4. Construct a lineAK that is perpendicular to AD , with K being the intersection with the line LM that is parallel to AD .
5. The rectangle◇ACLK has been constructed with the area equal to that of rectangle ◇HMFG .
6. The sum of the areas of rectangle◇ADMK (ￜAD ￜ⋅ￜBD ￜ ) and square ◇LHGE (ￜCB ￜ2 ) is equal to the area of square ◇CDFE (ￜCD ￜ2 ).
7. This geometric relationship is similar to the previous proposition, Book 2 Proposition 5, and can be expressed algebraically as(x + y) (x - y) + y2 = x2 (where ￜCD ￜ = x and ￜCB ￜ = y ). This algebraic relationship is one of the most used polynomial identities, referred to as the difference of two squares, and commonly expressed as x2 - y2 = (x + y) (x - y) .
2. Construct a square
3. Through H, draw a line parallel to
4. Construct a line
5. The rectangle
6. The sum of the areas of rectangle
7. This geometric relationship is similar to the previous proposition, Book 2 Proposition 5, and can be expressed algebraically as
Original statement
ἐὰν ϵὐθϵῖα γραμμὴ τμηθῇ δίχα, προστϵθῇ δέ τις αὐτῇ ϵὐθϵῖα ἐπ᾽ ϵὐθϵίας, τὸ ὑπὸ τῆς ὅλης σὺν τῇ προσκϵιμένῃ καὶ τῆς προσκϵιμένης πϵριϵχόμϵνον ὀρθογώνιον μϵτὰ τοῦ ἀπὸ τῆς ἡμισϵίας τϵτραγώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς συγκϵιμένης ἔκ τϵ τῆς ἡμισϵίας καὶ τῆς προσκϵιμένης τϵτραγώνῳ.
English translation
If a straight line is bisected and a straight line is added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line.
