construct perpendicular from point on line

I.11

To draw a straight line at right angles to a given straight line from a given point on it.

—Euclid

Let AB be the given straight line, and C the given point on it.

PROBLEM: draw from the point C a straight line at right angles to the straight line AB.

Let a point D be taken at random on AC; [1] let CE be made equal to CD; [I.3] on DE let the equilateral triangle FDE be constructed, [I.1] and let FC be joined;

I say that the straight line FC has been drawn at right angles to the given straight line AB from C the given point on it.

For, since DC is equal to CE, and CF is common,

• the two sides DC, CF are equal to the two sides EC, CF respectively;

and the base DF is equal to the base FE;

• therefore the angle DCF is equal to the angle ECF; [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; [I.def.10]

• therefore each of the angles DCF, FCE is right.

Therefore the straight line CF has been drawn at right angles to the given straight line AB from the given point C on it.

references

[I.def.10]: /elem.1.def.10 “Book I - Definition 10” [I.1]: /elem.1.1 “Book I - Proposition 1” [I.3]: /elem.1.3 “Book I - Proposition 3” [I.8]: /elem.1.8 “Book I - Proposition 8”

footnotes

[1]

let CE be made equal to CD. The verb is <foreign lang=”greek”>κείσθω</foreign> which, as well as the other parts of <foreign lang=”greek”>κεῖμαι</foreign>, is constantly used for the passive of <foreign lang=”greek”>τίθημι</foreign> <quote>to <em>place</em></quote> ; and the latter word is constantly used in the sense of <em>making</em>, e.g., one straight line equal to another straight line.