Kitap 3
10
A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.
11
11
Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.
BOOK III. PROPOSITIONS.
PROPOSITION 1.
1
To find the centre of a given circle.
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Let ABC be the given circle; thus it is required to find the centre of the circle ABC.
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Let a straight line AB be drawn through it at random, and let it be bisected at the point D; from D let DC be drawn at right angles to AB and let it be drawn through to E; let CE be bisected at F; I say that F is the centre of the circle ABC.
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For suppose it is not, but, if possible, let G be the centre, and let GA, GD, GB be joined.
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Then, since AD is equal to DB, and DG is common, the two sides AD, DG are equal to the two sides BD, DG respectively; and the base GA is equal to the base GB, for they are radii; therefore the angle ADG is equal to the angle GDB. [I. 8]
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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 the angle GDB is right.
1
But the angle FDB is also right; therefore the angle FDB is equal to the angle GDB, the greater to the less: which is impossible. Therefore G is not the centre of the circle ABC.
