Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/262

238 PROPOSITION 17. THEOREM.

If two straight lines be cut by parallel planes, they shall be cut in the same ratio.

Let the straight lines AB and CD be cut by the parallel planes GH, KL, MN, at the points A, E, B, and C,F,D: AE shall be to EB as CF is to FD.

Join AC,BD,AD; let AD meet the plane KL at the point X; and join EX, XF.

Then, because the two parallel planes KL, MN are cut by the plane EBDX, the common sections EX, BD are parallel; [XI. 16. and because the two parallel planes GH, KL are cut by the plane AXFC, the common sections AC, XF are parallel. [XI. 16.

And, because EX is parallel to BD, a side of the triangle ABD, therefore AE is to EB as AX is to XD. [VI. 2.

Again, because XF is parallel to AC, a side of the triangle ADC, therefore AX is to XD as CF is to FD. [VI. 2. And it was shewn that AX is to XD as AE is to EB; therefore AE is to EB as CF is to FD. [V. 11.

Wherefore, if two straight lines &c.

PROPOSITION 18. THEOREM.

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.