SECTION IV.

ANGLES IN SEGMENTS.

DEF. 9. A segment of a circle is the figure contained by a chord and either of the arcs into which the chord divides the circumference. The segments are called major or minor segments according as the arcs that bound them are major or minor arcs.

DEF. 10. The angle formed by any two chords drawn from a point on the circumference of a circle is called an angle at the circumference, and is said to stand upon the arc between its arms.

DEF. II. An angle contained by two straight lines drawn from a point in the arc of a segment to the extremities of the chord is called an angle in the segment.

THEOR. 15. An angle at the circumference is half the angle at the centre standing on the same arc.

Let BAC be at an angle at the circumference, BOC an angle at the centre standing on the same arc BEC:

then shall the angle BAC be half the angle BOC.

First let the centre O lie on AB, one of the arms of the angle BAC.

B

Then, because OA is equal to OC,

III. Def. 1. 1. 7.

therefore the angle OCA is equal to the angle OAC; therefore the angle OAC is equal to half the sum of the angles OAC, OCA;

but the angle BOC, being the exterior angle of the triangle OAC, is equal to the sum of the angles OAC, OCA; I. 25. therefore the angle BAC is half the angle BOC, standing on the arc BEC.

Next, let O lie within the angle BAC.

Join AO, and produce AO to meet the circumference at D.

Then, as before, the angle BAO is half the angle BOD,

and the angle OAC is half the angle DOC;

therefore the whole angle BAC is half the whole angle BOC standing on the arc BEC.

Again, let O lie without the angle BAC :

B

As before, join AO, and produce AO to meet the circumference at D,

then, the angle BAO being half the angle BOD, and the angle OAC half the angle DOC,

therefore the remaining angle BAC is half the remaining angle BOC.

Q.E.D.

* Ex. 41. Two chords AB, CD intersect at a point E within the circle. Shew that the angle AEC is half the sum of the angles at the centre standing on the arcs AC, BD.

* Ex, 42. Two chords AB, CD, produced intersect at a point E without the circle. Shew that the angle AEC is half the difference of the angles at the centre standing on the arcs AC, BD.

THEOR. 16. Angles in the same segment are equal to one

another.

Let BAC, BDC be angles in the same segment BADC:

then shall the angle BAC be equal to the angle BDC.

Take O the centre of the circle; join OB, QC.

Then the angles BAC, BDC are angles standing on the same arc, namely the arc BEC which is conjugate to the arc BADC, and therefore each is half of the angle BOC on the arc BEC: III. 15. therefore the angle BAC is equal to the angle BDC.

Ax. h. Q.E.D.

COR. I. The angle subtended by the chord of a segment at a point within it is greater than, and that at a point outside the segment and on the same side of the chord as the segment is less than, the angle in the segment.

Let D be a point within the segment BAC :

then shall the angle BDC be greater than the angle in tne segment BAC.

Produce BD to meet the circumference at E, and join EC.

I. 9.

Then, the exterior angle BDC of the triangle DEC is greater than the interior opposite angle DEC; that is, the angle BDC is greater than the angle BEC which is in the segment ABC.

In like manner it may be shown that if D is a point outside the segment ABC and on the same side of the chord BC as the segment; then the angle BDC is less than the angle in the segment BAC.

COR. 2. The locus of a point on one side of a given finite straight line at which that line subtends a constant angle is an arc of which that line is the chord.

* Ex. 43. Two chords AB, CD of a circle intersect in E. Shew that whether E be within or without the circle the triangles AEC, DEB are equiangular. ECB.

So also are the triangles EAD,

Ex. 44. BA is a chord of one, BC of the other, of two circles which intersect at B. Through B any straight line PBQ is drawn meeting the circumferences again in P and Q respectively. Shew that if PA, QC, produced if necessary, meet in R, R lies on a certain circular arc.

Ex. 45. P is any point on a circular arc APB, AP is produced to Q, so that PQ is equal to PB: shew that the locus of Q is a circular arc,

THEOR. 17. The angle in a segment is greater than, equal to, or less than, a right angle, according as the segment is less than, equal to, or greater than, a semicircle.