Book 1
10
For, since AC is equal to CB, and CD is common, the two sides AC, CD are equal to the two sides BC, CD respectively; and the angle ACD is equal to the angle BCD; therefore the base AD is equal to the base BD. [I. 4]
10
Therefore the given finite straight line AB has been bisected at D.
10
Q. E. F.
Proposition 11.
11
Enunciation To draw a straight line at right angles to a given straight line from a given point on it.
11
Proof. Let AB be the given straight line, and C the given point on it.
11
Thus it is required to draw from the point C a straight line at right angles to the straight line AB.
11
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;
11
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.
11
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.
11
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.
11
Therefore the straight line CF has been drawn at right angles to the given straight line AB from the given point C on it.
11
Q. E. F.
Proposition 12.
12
Enunciation To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.
12
Proof. 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.
