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II. The portion AC of the secant PQ, intercepted by the circle, is called a CHORD.
III. The two portions, into which a chord divides the circumference, as A BC and ADC, are called ARCS.
IV. The two figures into which a chord divides the circle, as A BC and ADC, that is, the figures, of which the boundaries are respectively the arc ABC and the chord AC, and the arc ADC and the chord AC, are called SEGMENTS of the circle.
The figure AOCD, whose boundaries are two radii and the arc intercepted by them, is called a SECTOR.
VI. A circle is said to be described about a rectilinear figure; when the circumference passes through
each of the angular points of the figure; and the figure is said to be inscribed in the circle.
PROPOSITION I. THEOREM.
The line, which bisects a chord of a circle at right angles, must contain the centre.
Let ABC be the given O. Let the st. line CE bisect the chord AB at rt. angles in D.
Then the centre of the O must lie in CE. For if not, let 0, a pt. out of CE, be the centre, and join OA, OD, OB.
Then, in As ODA, ODB,
1. C. .: LODB is a right ..
1. Def. 9. But _CDB is a right L, by construction; .:: LODB= LCDB, which is impossible ;
..O is not the centre.
Thus it may be shewn that no point, out of CE, can be the centre; and .. the centre must lie in CE.
COR. If CE be bisected in F, then F is the centre of the circle.
PROPOSITION II. THEOREM.
If any two points be taken in the circumference of a circle, the straight line, which joins them, must fall within the circle.
Let A and B be any two pts. in the Oce of the O ABC.
Then must the st. line AB fall within the O.
and : OB is greater than OD.
I. 19, :: the distance of D from O is less than the radius of the o,
and :: D lies within the O.
And the same may be shewn of any other pt. in AB.
Q. E. D. PROPOSITION III. THEOREM.
If a straight line, drawn through the centre of a circle, bisect a chord of the circle, which does not pass through the centre, it must cut it at right angles: and conversely, if it cut it at right angles, it must bisect it.
In the O ABC, let the chord AB, which does not pass through the centre 0, be bisected in E by the diameter CD.
Then must CD be I to AB.
Join OA, OB.
Then in As AEO, BEO,
.. LOEAN LOEB.
1. Def. 9.
Then must CD bisect AB.
1. E. Cor. p. 43. that is, CD bisects AB.
Q. E. D. Ex. 1. Shew that CD does not cut AB at right angles, it cannot bisect it.
Ex. 2. A line, which bisects two parallel chords in a circle, is also perpendicular to them.
Ex. 3. Through a given point within a circle, which is not the centre, draw a chord which shall be bisected in that point.
PROPOSITION IV. THEOREM.
If in a circle two chords, which do not both pass through the centre, cut one another, they do not bisect each other.
Let the chords AB, CD, which do not both pass through the centre, cut one another, in the pt. E, in the O ACBD.
Then AB, CD do not bisect each other. If one of them pass through the centre, it is plainly not bisected by the other, which does not pass through the centre.
But if neither pass through the centre, find the centre 0, and join OE.
Then : OE, passing through the centre, bisects AB,
.. LOEA is a rt. L.
And :: OE, passing through the centre, bisects CD,
:: LOEC is a rt. L;
:: ZOEA= LOEC, which is impossible ;
.. AB, CD do not bisect each other.
Q. E. D.
Ex. 1. Shew that the locus of the points of bisection of all parallel chords of a circle is a straight line.
Ex.2. Shew that no parallelogram, except those which are rectangular, can be inscribed in a circle.