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Proposition 11.

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

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

Thus it is required to 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;
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; [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.

Q. E. F.


1 The verb is κείσθω which, as well as the other parts of κεῖμαι, is constantly used for the passive of τίθημι “to place” ; and the latter word is constantly used in the sense of making, e.g., one straight line equal to another straight line.

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