Kitap 12
17
Therefore there are as many straight lines in the quadrants BO, KO equal to the straight lines BK, KL, LM, ME as there are sides of the polygon in the quadrant BE.
17
Let them be inscribed, and let them be BP, PQ, QR, RO and KS, ST, TU, UO, let SP, TQ, UR be joined, and from P, S let perpendiculars be drawn to the plane of the circle BCDE; [XI. 11] these will fall on BD, KN, the common sections of the planes, inasmuch as the planes of BOD, KON are also at right angles to the plane of the circle BCDE. [cf. XI. Def. 4]
17
Let them so fall, and let them be PV, SW, and let WV be joined.
17
Now since, in the equal semicircles BOD, KON, equal straight lines BP, KS have been cut off, and the perpendiculars PV, SW have been drawn, therefore PV is equal to SW, and BV to KW. [III. 27, I. 26]
17
But the whole BA is also equal to the whole KA; therefore the remainder VA is also equal to the remainder WA; therefore, as BV is to VA, so is KW to WA; therefore WV is parallel to KB. [VI. 2]
17
And, since each of the straight lines PV, SW is at right angles to the plane of the circle BCDE, therefore PV is parallel to SW. [XI. 6]
17
But it was also proved equal to it; therefore WV, SP are also equal and parallel. [I. 33]
17
And, since WV is parallel to SP, while WV is parallel to KB, therefore SP is also parallel to KB. [XI. 9]
17
And BP, KS join their extremities; therefore the quadrilateral KBPS is in one plane, inasmuch as, if two straight lines be parallel, and points be taken at random on each of them, the straight line joining the points is in the same plane with the parallels. [XI. 7]
17
For the same reason each of the quadrilaterals SPQT, TQRU is also in one plane.
17
But the triangle URO is also in one plane. [XI. 2]
