(ε). II. 8.
Through a given point, without a given Line, a Line may be drawn such that the two Lines are equidistantial from each other.
For, if through the given point there be drawn a transversal, there can also be drawn through it a Line such that the two Lines make equal ∠s with the transversal; [Euc. I. 23.
and this Line will be such that the two Lines are equidistantial from each other. [(δ).
Therefore, &c. Q. E. D.
(ζ). II. 17.
A Line cannot recede from and then approach another; nor can one approach and then recede from another on the same side of it.
If possible, let ABC first recede from, and then approach, DE; that is, let the perpendicular BG be > each of the two perpendiculars AF, CH.
From GB cut off GK > each of the two, AF, CH.
Now a Line may be drawn, through K, equidistantial from DE; [(ε).
and the points A, C will lie on the side of it next to DE, and B on the other side;
∴ it will cut AB between A and B, and BC between B and C.
Let L, M be the points of intersection; and join LM;
∴ the 2 Lines LBM, LKM contain a space; which is absurd.
Similarly it may be proved that ABC cannot first approach and then recede from DE on the same side of it.
Therefore a Line &c. Q. E. D.
