Elements of geometry and trigonometry . ibed angle, and letus first suppose that the centre of the cir-cle lies within the angle BAD. Draw thediameter AE, and the radii CB, CD. The angle BCE, being exterior to thetriangle ABC, is equal to the sum of thetwo interior angles CAB, ABC (Book XXV. Cor. 6.) : but the triangle BACbeing isosceles, the angle CAB is equal toABC : hence the angle BCE is double of BAC. BCE liesat the centre, it is measured by the arc BE ; hence BAC will bemeasured by the half of BE. For a like reason, the angle CADwill be measured by the half of ED ; hence B
Elements of geometry and trigonometry . ibed angle, and letus first suppose that the centre of the cir-cle lies within the angle BAD. Draw thediameter AE, and the radii CB, CD. The angle BCE, being exterior to thetriangle ABC, is equal to the sum of thetwo interior angles CAB, ABC (Book XXV. Cor. 6.) : but the triangle BACbeing isosceles, the angle CAB is equal toABC : hence the angle BCE is double of BAC. BCE liesat the centre, it is measured by the arc BE ; hence BAC will bemeasured by the half of BE. For a like reason, the angle CADwill be measured by the half of ED ; hence BAC + CAD, or BADwill be measured by half of BE + ED, or of BED. Suppose, in the second place, that thecentre C lies without the angle BAD. Thendrawing the diameter AE, the angle BAEwill be measured by the half of BE ; theangle DAE by the half of DE : hence theirdifference BAD will be measured by thehalf of BE minus the half of ED, or by thehalf of BD. Hence every inscribed angle is measuredbv half of the arc included between its BOOK III. 55 Cor. 1. All the angles BAC, BDC,BEC, inscribed in the same segment areequal ; because they are all measured bythe half of the same arc BOC. Cor, 2. Every angle BAD, inscribed in asemicircle is a right angle ; because it is mea-sured by half the semicircumferencc BOD,that is, by the fourth part of the ^vilole cir-cumference. Cor. 3. Ever} angle BAC, inscribed in asegment greater than a semicircle, is an acuteangle ; for it is measured by half of the arcBOC, less than a semicircumferencc. And every angle BOC, inscribed in ase^^ment less than a semicircle, is an obtuseangle ; for it is measured by half of the arc ^BAC, greater than a semicircumference. Cor. 4. Tlie opposite angles A and C, ofan inscribed quadrilateral ABCD, are to-gether equal to two right angles : for the an-gle BAD is measured by half the arc BCD,the angle BCD is measured by half the arcBAD ; hence the two angles BAD, BCD, ta-ken together, are measured by the half of
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