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THIRD AND FOURTH BOOKS OF EUCLID
EXERCISES AND NOTES
J. HAMBLIN SMITH M. A
GONVILLE AND CAIUS COLLEGE, AND LATE LECTURER AT ST PETER'S COLLEGE,
A POINT is within, or without, a circle, according as its distance from the centre is less, or greater than, the radius of the circle.
I. A straight line, as PQ, drawn so as to cut a circle ABCD, is called a SECANT.
That such a line can only meet the circumference in two points may be shewn thus:
Some point within the circle is the centre; let this be 0. Join OA. Then (Ex. 1, 1. 16) we can draw one, and only one, straight line from O, to meet the straight line PQ, such that it shall be equal to OA. Let this line be OC. Then A and Care the only points in PQ, which are on the circumference of the circle.
S. E. II.
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 ABC and ADC, are called ARCS.
IV. The two figures into which a chord divides the circle, as ABC 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.
V. 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 the st. line CE bisect the chord AB at rt. angles in D.
Then the centre of the
must lie in CE.
For if not, let O, a pt. out of CE, be the centre,
and join OA, OD, OB.
Then, in AS ODA, ODB,
:: AD=BD, and DO is common, and OA=0B;
:. LODA= LODB;
.. 4 ODB is a right 4.
But CDB is a right, by construction;
.. LODB= LCDB, which is impossible;
.. O is not the centre.
I. Def. 9.
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.