Commentary on Proposition II.8¶
Back to Proposition II.8.
This proposition is quoted by Pappus (p. 428, ed. Hultsch) and is used also by Euclid himself in the Data, Prop. 86. Further, it is of decided use in proving the fundamental property of a parabola.
Two alternative proofs are worth giving.
The first is that suggested by the consideration mentioned in the last note, though the proof is old enough, being given by Clavius and others. It is of the semi-algebraical type.
Produce AB to D (in the figure of the proposition), so that BD is equal to BC.
By II. 4, the square on AD is equal to the squares on AB, BD and twice the rectangle AB, BD, i.e. to the squares on AB, BC and twice the rectangle AB, BC.
By II. 7, the squares on AB, BC are equal to twice the rectangle AB, BC together with the square on AC.
Therefore the square on AD is equal to four times the rectangle AB, BC together with the square on AC.
The second proof is after the manner of Euclid but with a difference. Produce BA to D so that AD is equal to BC. On BD construct the square BEFD.
Take BG, EH, FK each equal to BC or AD, and draw ALP, HNM parallel to BE and GML, KPN parallel to BD.
Then it can be shown that each of the rectangles BL, AK, FN, EM is equal to the rectangle AB, BC, and that PM is equal to the square on AC.
Therefore the square on BD is equal to four times the rectangle AB, BC together with the square on AC.