Kitap 11
4
And, since AD is equal to CB, and FA is also equal to FB, the two sides FA, AD are equal to the two sides FB, BC respectively; and the base FD was proved equal to the base FC; therefore the angle FAD is also equal to the angle FBC. [I. 8]
4
And since, again, AG was proved equal to BH, and further FA also equal to FB, the two sides FA, AG are equal to the two sides FB, BH.
4
And the angle FAG was proved equal to the angle FBH; therefore the base FG is equal to the base FH. [I. 4]
4
Now since, again, GE was proved equal to EH, and EF is common, the two sides GE, EF are equal to the two sides HE, EF; and the base FG is equal to the base FH; therefore the angle GEF is equal to the angle HEF. [I. 8]
4
Therefore each of the angles GEF, HEF is right.
4
Therefore FE is at right angles to GH drawn at random through E.
4
Similarly we can prove that FE will also make right angles with all the straight lines which meet it and are in the plane of reference.
4
But a straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in that same plane; [XI. Def. 3] therefore FE is at right angles to the plane of reference.
4
But the plane of reference is the plane through the straight lines AB, CD.
4
Therefore FE is at right angles to the plane through AB, CD.
4
Therefore etc. Q. E. D.
PROPOSITION 5.
5
If a straight line be set up at right angles to three straight lines which meet one another, at their common point of section, the three straight lines are in one plane.
