Proposition 12

To draw a straight line perpendicular to a given infinite straight line from a given point not on it.

Let AB be the given infinite straight line, and C the given point which is not on it.

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It is required to draw a straight line perpendicular to the given infinite straight line AB from the given point C which is not on it.

I.Post.3
I.10, I.Post.1

Take an arbitrary point D on the other side of the straight line AB, and describe the circle EFG with center C and radius CD. Bisect the straight line EG at H, and join the straight lines CG, CH, and CE.

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.

I.Def.15
I.8

Since GH equals HE, and HC is common, therefore the two sides GH and HC equal the two sides EH and HC respectively, and the base CG equals the base CE. Therefore the angle CHG equals the angle EHC, and they are adjacent angles.

I.Def.10

But, when a straight line standing 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.

Therefore CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.

Q.E.F.

Guide

 Again, the double-equilateral-triangle construction is used, but this time the preparation of the starting line EG is different. The point D is taken on the other side of the line AB to insure that circle meets the line AB in at least two points, E and G. If D is taken on the line AB, it might be taken at H, and the resulting circle would touch the line only at H; and if D is taken on the same side of AB, then the circle could miss the line entirely.

Euclid does not precede this proposition with propositions investigating how lines meet circles. He is much more careful in Book III on circles in which the first dozen or so propositions lay foundations. For instance, Proposition III.10 states that a circle does not cut a circle at more than two points. Even so, some propositions are missing. One is needed for this proposition to justify the existence of the two points C and E where the line AB meets circle with center C and radius CD. Such a proposition would state “A circle whose center is on one side of a line and on whose circumference lies a point on the other side of the line meets the line at two points.”

Incidentally, Proclus explains in his commentary on Book I that the problem of constructing the perpendicular was investigated by Oenopides of Chios who lived sometime in the middle of the fifth century B.C.E., a century and a half before Euclid.

Use of Proposition 12

The construction of this proposition is not used in Book I, but it is used on occasion in the remaining geometric books, namely, Books II through IV, VI, and XI through XIII.