Proposition 10
Theorem
If a line AB is bisected at point C, and divided externally at point D, then ￜAD ￜ2 + ￜBD ￜ2 = 2 ￜAC ￜ2 + 2 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 lineCE that is perpendicular to AB and has the same length as AC . Connect AE and EB .
3. ExtendEB to a point F such that ◇CDFG is a rectangle with G on the extension of EC . Connect AF .
4. Five isosceles right triangles are constructed:△ACE , △BCE , △AEB , △EGF and △FDB . Two right triangles that are not isosceles are constructed: △ADF and △AEF .
5. Apply the Pythagorean Theorem to the right triangles so: in△ADF , ￜAF ￜ2 = ￜAD ￜ2 + ￜDF ￜ2 , in △AEF , ￜAF ￜ2 = ￜAE ￜ2 + ￜEF ￜ2 , in △ACE , ￜAE ￜ2 = ￜAC ￜ2 + ￜCE ￜ2 and in △EGF , ￜEF ￜ2 = ￜEG ￜ2 + ￜGF ￜ2 . With substitution of lines with the same length, the following equation is obtained: ￜAD ￜ2 + ￜBD ￜ2 = 2 ￜAC ￜ2 + 2 ￜCD ￜ2 .
6. This geometric relationship is similar to the previous proposition, Book 2 Proposition 9, and can also be expressed algebraically asx + y2 + x - y2 = 2 x2 + 2 y2 (where ￜAC ￜ = ￜCE ￜ = x and ￜCD ￜ = y ). This algebraic relationship is an interpretation of a commonly used polynomial identity.
2. Construct a line
3. Extend
4. Five isosceles right triangles are constructed:
5. Apply the Pythagorean Theorem to the right triangles so: in
6. This geometric relationship is similar to the previous proposition, Book 2 Proposition 9, and can also be expressed algebraically as
Original statement
ἐὰν ϵὐθϵῖα γραμμὴ τμηθῇ δίχα, προστϵθῇ δέ τις αὐτῇ ϵὐθϵῖα ἐπ᾽ ϵὐθϵίας, τὸ ἀπὸ τῆς ὅλης σὺν τῇ προσκϵιμένῃ καὶ τὸ ἀπὸ τῆς προσκϵιμένης τὰ συναμϕότϵρα τϵτράγωνα διπλάσιά ἐστι τοῦ τϵ ἀπὸ τῆς ἡμισϵίας καὶ τοῦ ἀπὸ τῆς συγκϵιμένης ἔκ τϵ τῆς ἡμισϵίας καὶ τῆς προσκϵιμένης ὡς ἀπὸ μιᾶς ἀναγραϕέντος τϵτραγώνου.
English translation
If a straight line is bisected, and a straight line is added to it in a straight line, the square on the whole with the added straight line and the square on the added straight line both together are double of the square on the half and of the square described on the straight line made up of the half and the added straight line as on one straight line.
