To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.
给定无限直线 AB 及其外一点 C，从 C 作垂线到 AB。
C 在 AB 外；D 在 AB 异侧；以 C 为心 CD 为半径作圆 EFG，交 AB 于 E、G；H 是 EG 中点，CH 垂直 AB。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let AB be the given infinite straight line, and C the given point which is not on it; thus it is required to draw to the given infinite straight line AB, from the given point C which is not on it, a perpendicular straight line. For let a point D be taken at random on the other side of the straight line AB, and with centre C and distance CD let the circle EFG be described; [Post. 3] let the straight line EG be bisected at H, [I. 10] and let the straight lines CG, CH, CE be joined.
在给定直线 AB 上任取两点 D、E，使待作垂线的外点 C 与它们能构成辅助三角形。
[Post. 1] I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it. For, since GH is equal to HE, and HC is common, the two sides GH, HC are equal to the two sides EH, HC respectively; and the base CG is equal to the base CE; therefore the angle CHG is equal to the angle EHC.
按 euclid-elements/book1-prop-010 平分 DE，再连接 C 到平分点。
[I. 8] And they are adjacent angles. But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
利用两侧相等线段和 euclid-elements/book1-prop-008，得到平分点处相邻角相等。
[Def. 10] Therefore CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.
相邻角又在同一直线上，所以它们是直角，从 C 作得垂线。