In the plane, draw any straight line BC, and from the point, A, draw * AD perpendicular to BC: if then AD be also per- . 12. 1, pendicular to the plane BH, the thing required is already done: but if it be not, from the point D, draw *, in the plane BH, • 11. 1. the straight line DE at right angles to BC; and from the point A, draw AF perpendicular to DE: AF shall be perpendicular to the plane BH. Through F, draw GH parallel to BC: and because BC is • 31. 1. at right angles to ED and DA, BC is at right angles to the * 4. 11. plane passing through ED, DA; and GH is parallel to BC: but, if two straight lines be parallel, one of which is at right angles to a plane, the other is at right * angles to the same • 8. II. plane; wherefore GH is at right angles to the plane through ED, DA; and is perpendicular* to every straight line meet- • 3 Def. 11. ing it in that plane: but AF, which is in the plane through ED, DA, meets it; therefore GH is perpendicular to AF; and consequently AF is perpendicular to GH: and AF is perpendicular to DE; therefore AF is perpendicular to each of the straight lines GH, DE. But if a straight line stand at right angles to each of two straight lines in the point of their intersection, it is also at right angles † to the plane passing through † 4. 11. them: but the plane passing through ED, GH, is the plane BH ; therefore AF is perpendicular to the plane BH: therefore, from the given point A, above the plane BH, the straight line AF is drawn perpendicular to that planc. Which was to be done. PROPOSITION XII. PROB.To erect a straight line at right angles to a given plane, from a point given in the plane. Let A be the point given in the plane; it is required to erect a straight line from the point A, at right angles to the plane. From any point B above the plane, draw * D B BC perpendicular to it; and from A, draw AD parallel to BC. Because, therefore, AD, CB are two parallel straight lines, and one of them BC is at right angles to the given plane, the other ad is also * at right angles to it: therefore, a . ll. 11. * 31. 1. 8. IR. straight line has been erected at right angles to a given plane, from a point given in it. Which was to be done. PROPOSITION XIII. THEOR.–From the same point in a given plane, there cannot be two straight lines at right angles to the plane, upon the same side of it: and there can be but one perpendicular to a plane from a point above the plane. For, if it be possible, let the two straight lines AB, AC be at right angles to a given plane, from the same point A in the plane, and upon the same side of it. Let a plane pass through BA, AC; the common section of this with the given plane is * 3. 11. a straight line passing through A: let B A E + 3 Def. 11. given plane, it makes right angles † with every straight line meeting it in that plane: but DAE, which is in that plane, meets CA ; therefore CAE is a right angle: for the same reason BAE is a right angle; wherefore the angle + 11 Ax. CAE is equal t to the angle BAE ; and they are in one plane, which is impossible. Also, from a point above a plane, there can be but one perpendicular to that plane; for if there could • 6. 11. be two, they would be parallel * to one another, which is ab surd. Therefore, from the same point, &c. Q. E. D. PROPOSITION XIV. Theor.- Planes to which the same straight line is perpen dicular, are parallel to one another. G C С E that plane; therefore ABK is a right angle: for the same reason BAK is a right angle ; wherefore the two angles ABK, BAK of the triangle ABK, are equal to two right angles; which is * impossible: therefore the planes CD, EF, though produced, * 17. 1. do not meet one another; that is *, they are parallel. There- * 8 Def. 11. fore, planes, &c. Q. E. D. PROPOSITION XV. Theor.-If two straight lines meeting one another, be parallel Sce N. to Iwo other straight lines which meet one another but are not in the same plane with the first two, the plane which passes through these is parallel to the plane passing through the others. Let AB, BC, two straight lines meeting one another, be parallel to DE, EF that meet one another, but are not in the same plane with AB, BC: the planes through AB, BC, and DE, EF shall not meet, though produced. From the point B, draw BG perpendicular * to the plane 11. 11. which passes through DE, EF, and let it meet that plane in G; and through G, draw GH parallel * to • 31. 1. ED, and GK parallel to EF. And because E. K * 3 Def. 11. straight line meeting it in that plane: but AY the straight lines GH, GK in that plane meet it; therefore each of the angles BGH, BGK is a right angle: and because BA is parallel * to GH, (for each of them. 9. 11. is parallel to DE, and they are not both in the same plane with it,) the angles GBA, BGH are together equal * to two right angles: and BGH is a right angle; therefore also GBA is a right angle, and GB perpendicular to BA: for the same reason GB is perpendicular to BC; since therefore the straight line GB stands at right angles to the two straight lines BA, BC that cut one another in B, GB is perpendicular * to the plane • 4. 11. through BA, BC: and it is perpendicular † to the plane through + Constr. DE, EF; therefore BG is perpendicular to each of the planes through AB, BC, and DE, EF: but planes to which the same straight line is perpendicular, are parallel * to one another; . 14. 11. therefore the plane through AB, BC is parallel to the plane through DE, EF. Wherefore, if two straight lines, &c. Q. E. D. * 29. I. PROPOSITION XVI. THEOR.-If two parallel planes be cut by another plane, their common sections with it are parallels. Let the parallel planes AB, CD be cut by the plane EFHG, and let their common sections with it be EF, GH: EF shall be parallel to GH. For if it is not, EF, GH shall meet if produced either on K H B plane CD; wherefore the planes AB, CD, produced meet one another: but they do not meet, since they are parallel by the hypothesis ; therefore the straight lines EF, GH do not meet when produced on the side of FH: in the same manner it may be proved, that EF, GH do not meet when produced on the side of EG. But straight lines which are in the same plane, and do not meet, though produced either way, are parallel ; therefore EF is parallel to GH. Wherefore, if two parallel planes, &c. Q. E. D. PROPOSITION XVII. Theor.-If two straight lines be cut by parallel planes, they shall be cut in the same ratio. C Let the straight lines AB, CD be cut by the parallel planes н Join AC, BD, AD, and let AD meet the G L E MN are cut by the plane EBDX, the com- KY mon sections EX, BD are * parallel : for N M4 the same reason, because the two parallel planes GH, KL are cut by the plane AXFC, the common tions AC, XF are parallel : and because Ex is parallel to BD, # 16. 11. B sec a side of the triangle ABD; as AE to EB, so is * AX to XD: * 2. 6. again, because XF is parallel to AC, a side of the triangle ADC; as AX to XD, so is CF to FD: and it was proved, that AX is to XD, as AE to EB; therefore *, as AE to EB, so is CF to + 11.5. FD. Wherefore, if two straight lines, &c. Q. E. D. PROPOSITION XVIII. THEOR.-If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane. Let the straight line AB be at right angles to the plane CK: every plane which passes through AB, shall be at right angles to the plane CK. Let any plane DE pass through AB, and let CE be the common section of the planes DE, CK; take any point F in CE, from which draw FG, in the plane DE, at right + angles to CE: and because AB is 11. 1. ІІ perpendicular to the plane CK, therefore * * 3 Def. 11. it is also perpendicular to every straight K line in that plane meeting it, and consequently it is perpendicular to CE; where F B B fore ABF is a right angle: but GFB is likewise + a right angle; therefore AB is parallel * to FG: and AB + Constr. is at right angles to the plane CK; therefore FG is also at right angles to the same plane. But one plane is at right angles to another plane, when the straight lines drawn in one of the planes at right angles to their common section, are also at right angles to the other plane; and any straight line FG • 4 Def. 11. in the plane DE, which is at right angles to CE, the common section of the planes, has been proved to be perpendicular to the other plane CK; therefore the plane DE is at right angles to the plane CK. In like manner it may be proved, that all planes which pass through AB, are at right angles to the plane CK. Therefore, if a straight line, &c. E * • 28. 1. * 8. 11. Q. E. D. PROPOSITION XIX. Theor.-If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane. |