Kitap 11
2
But, in whatever plane the triangle ECB is, in that plane also is each of the straight lines EC, EB, and, in whatever plane each of the straight lines EC, EB is, in that plane are AB, CD also. [XI. 1]
2
Therefore the straight lines AB, CD are in one plane, and every triangle is in one plane. Q. E. D.
PROPOSITION 3.
3
If two planes cut one another, their common section is a straight line.
3
For let the two planes AB, BC cut one another, and let the line DB be their common section; I say that the line DB is a straight line.
3
For, if not, from D to B let the straight line DEB be joined in the plane AB, and in the plane BC the straight line DFB.
3
Then the two straight lines DEB, DFB will have the same extremities, and will clearly enclose an area: which is absurd.
3
Therefore DEB, DFB are not straight lines.
3
Similarly we can prove that neither will there be any other straight line joined from D to B except DB the common section of the planes AB, BC.
3
Therefore etc. Q. E. D.
PROPOSITION 4.
4
If a straight line be set up at right angles to two straight lines which cut one another, at their common point of section, it will also be at right angles to the plane through them.
4
For let a straight line EF be set up at right angles to the two straight lines AB, CD, which cut one another at the point E, from E; I say that EF is also at right angles to the plane through AB, CD.
4
For let AE, EB, CE, ED be cut off equal to one another, and let any straight line GEH be drawn across through E, at random; let AD, CB be joined, and further let FA, FG, FD, FC, FH, FB be joined from the point F taken at random lton EFgt.
4
Now, since the two straight lines AE, ED are equal to the two straight lines CE, EB, and contain equal angles, [I. 15] therefore the base AD is equal to the base CB, and the triangle AED will be equal to the triangle CEB; [I. 4] so that the angle DAE is also equal to the angle EBC.
