ABC and ABD are two straight lines, and they have the PROP. II. THEOR. ANY three straight lines which meet one another, but not in the same point, are in one plane. Let the three straight lines AB, CD, CB meet one another in the points B, C, E; they are in one plane. A E a Def. 3. 1. Let any plane pass through the straight line EB, and let the plane be turned about EB (produced, if necessary) until it pass through the point C; then because the points E, C are in this plane, the straight line EC is in ita. For the same reason BC is in the same plane; and, by the hypothesis, EB is in it. Therefore the three straight lines EC, CB, BE are in one plane. But the whole lines DC, AB, BC, produced, are in the same plane with the parts of them EC, EB, BC. Therefore AB, CD, B CB are all in one plane. Wherefore, any three straight lines, &c. Q. E. D. COR. It is manifest that any two straight lines which cut each other are in one plane; and that any three points whatever are in one plane. b 1. 2. Sup. PROP. III. THEOR. IF two planes cut each other, their common section is a straight line. Suppl. B Let two planes AB, BC cut each other, and let B, D be two points in the line of their common section. From B to D draw the straight line BD. Because the points B, D are in the plane AB, the straight à Def. 5. 1. line BD is in the plane ABa; and because the points B, D are in the plane BC, the straight line BD is in the plane BC; therefore the straight line BD is common to the planes AB and BC, or it is the common section of these planes. Therefore, if two planes, &c. Q. E. D., D A PROP. IV. THEOR. IF a straight line stand at right angles to each of two straight lines in the point of their intersection, it will also be at right angles to the plane in which these lines are. Let the straight line AB stand at right angles to each of the straight lines EF, CD in A, the point of their intersection; AB is also at right angles to the plane passing through EF, CD. Through A draw any line AG in the plane in which are EF, CD, and let G be any point in AG; draw GH parallel to AD, and make HF=HA; join FG, and produce it to meet CA in D; join BD, BG, BF. But Because GH is parallel to AD, H G D a 2. 6. AF2. BF2-AB2+AF2. Therefore BD2+ BF22AB2+ AD2+ Book II. Butb AD2+AF2--2AG2-2GF2; therefore BD2+ BF2=2AB2+2AG2+2GF2. But BD2+BF2b2BG2+2GF2; b A. 2. therefore 2BG2+ 2GF22AB2 + 2AG2+2GF2; therefore 2BG2-2AB2+2AG2, or BG2-AB2+AG2; wherefore BAG c 48. 1. is a right angle, therefore AB is perpendicular to AG. Now AG is any straight line drawn in the plane of the lines AD, AF; therefore AB is at right angles to the planed of the lines d Def. 1. AF, AD. Therefore, if a straight line, &c. Q. E. D. 2. Sup. PROP. V. THEOR. IF three straight lines meet in one point, and a straight line stand at right angles to each of them in that point, these three straight lines are in one and the same plane. Let the straight line AB stand at right angles to each of the three straight lines BC, BD, BE, in B, the point where they meet; BC, BD, BE are in one and the same plane. If not, let, if it be possible, BD and BE be in one plane, and BC above it; and let a plane pass through AB, BC; then the common section of this plane with the plane in which BD and BE are will be a straight linea. Let this line be BF; a 3.2. Sup. therefore the three straight lines AB, BC, BF are in one plane, which passes through AB, BC. Because AB stands at right A B C -F D E b 4. 2. Sup. Suppl. plane, which is impossible. Therefore the straight line BC is not above the plane in which are BD and BE; wherefore the three straight lines BC, BD, BE are in one and the same plane. Therefore, if three straight lines, &c. Q. E. D. a 48. 1. b 5. 2. Sup, c 2. 2. Sup. d 28. 1. PROP. VI. THEOR. (IF two straight lines be at right angles to the same plane they will be parallel to each other. Let the straight lines AB, CD be at right angles to the same plane BDE; AB is parallel to CD. Let them meet the plane in the points B, D. Draw DE at right angles to DB, in the plane BDE, and let E be any point in it. Join AE, AD, ÉB. Because AB is a right angle, AB2+BE2-AE2; and because BDE is a right angle, BE2BD24-DE2; therefore AB2+BD2+DE-AE2. But AB2+BD2 AD2; therefore AD2+ DE AE2, therefore ADE is a right anglea. Therefore ED is perpendicular to the three lines BD, DA, DC; whence these lines are in one plane?. But AB is in the plane in which are BD, DA; therefore AB, BD, DC are in one plane; and because ABD, BDC are right angles, AB is parallel to CDd. Wherefore, if two straight lines, &c. Q. E. D. A PROP. VII, THEOR. IF two straight lines be parallel, and one of them at right angles to a plane, the other is also at right angles to the same plane. Let AB, CD be two parallel straight lines, and let one of Book II. them AB be at right angles to a plane; the other CD is at right angles to the same plane. For, if CD be not per pendicular to the plane to PROP. VIII. THEOR. TWO straight lines which are parallel to the same straight line, though not both in the same plane with it, are parallel to each other. Let AB, CD be each of them parallel to EF, and not in the same plane with it; AB is parallel to CD. In EF take any point G, from which draw, in the plane passing through EF, AB, the straight line GH at right angles to EF; and in the H ·B right angles to EF. Because EF is perpendicular both to GH and GK, it is perpen dicular to the plane HGK passing through K them. But EF is pa rallel to AB; therefore AB is at right angles to the plane b 7.2. Sup. HGK. For the same reason CD is at right angles to the plane HGK. Therefore AB, CD are at right angles to the |