(η). II. 13.
A Pair of Lines, of which one has two points on the same side of, and equidistant from, the other, are eqiddistantial from each other.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/68/Euclid_and_His_Modern_Rivals_Page252.png/250px-Euclid_and_His_Modern_Rivals_Page252.png)
Let AB contain two points E, F, equidistant from CD. From E, F, draw EG, FH, ⊥ CD; bisect GH in K, and EF in L, and join KL.
Now EG = FH; [hyp.
hence, if the diagram be reversed, and so placed on its former traces that G coincides with H, and H with G, K retaining its position, GE coincides with HF, and HF with GE;
∴ E coincides with F, and F with E;
∴ L retains its position;
∴, if there be a point in LA whose distance is < LK, there is another such point in LB, and the Line AB will first recede from and then approach CD; which is absurd. [(ζ).
Similarly if there be one whose distance is > LK.
∴ AB is equidistantial from CD.
Similarly it may be proved that CD is equidistantial from AB.
Therefore a Pair of Lines, &c. Q. E. D.
Lemma 2.
Through a given point may he drawn a common perpendicular to a given Pair of Lines, of which each is equidistantial from the other.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/68/Euclid_and_His_Modern_Rivals_Page252.png/250px-Euclid_and_His_Modern_Rivals_Page252.png)