Elements of Geometry containing Books I to VI and portions of Books XI and XII of Euclid . In the ® AGBD, let the angles AOB (not less than tworight angles) at the centre, and ADB at the circumference, besubtended by the same arc ACE. Then must l AOB=ti(nce l DO, and produce it to meet the arc ACB in C. Then •.• z ^0<7=twice z ADO, III. 20. and z BOC= twice z BDO, III. 20. .. Bum of z s AOC, £0C=twice sum of z s ^ DO, BDO,that is, z J. 05=twice z ADB. Q. E. D. Note. In fig. 1, z AOB is drawn a flat angle,and in fig. 2, z ^ OB is drawn a reflex angle. Def. XII. The angle in a segmen
Elements of Geometry containing Books I to VI and portions of Books XI and XII of Euclid . In the ® AGBD, let the angles AOB (not less than tworight angles) at the centre, and ADB at the circumference, besubtended by the same arc ACE. Then must l AOB=ti(nce l DO, and produce it to meet the arc ACB in C. Then •.• z ^0<7=twice z ADO, III. 20. and z BOC= twice z BDO, III. 20. .. Bum of z s AOC, £0C=twice sum of z s ^ DO, BDO,that is, z J. 05=twice z ADB. Q. E. D. Note. In fig. 1, z AOB is drawn a flat angle,and in fig. 2, z ^ OB is drawn a reflex angle. Def. XII. The angle in a segment is the angle contained bytwo straight lines drawn from any point in the arc to the ex-tremities of the choFci- Bookm.] PROPOSITION XXI. 151 Proposition XXI. Theorek. Tht angles in the same segment of a circle are equal to on* Let BA Q, BDG be angles in the same segment BA DC. Then must L BAC= i. , when segment BADC is greater than a semicircle. From 0, the centre, draw OB, OC. (Fig. 1.) Then, •.• /. £0C= twice i BAG, III 20. and I BOO =twice L BBC, III. 20. .?.lBAC= I , when segment BADC is less than a semicircle, Let E be the pt. of intersection of AC, DB. (Fig. 2.)Then :? l ABE= l DCS, by the first case, &iid/.BEA= L CED, I. 15. .-. L EAB= L EDC, I. 32. that is, ^ BAC= i BDC. q. e. d. Ex. 1. Shew that, by assuming the possibility of an anglebeing greater than two right angles, both the cases of thisproposition may be included in one. Ex. 2. AB, AC are chords of a circle, D, E the middlepoints of their arcs. If DE be joined, shew that it will cutofiF equal parts from AB, A C. Ex. 3. If two straight lines, whose extremities are in thecircumference of a circle, cut one another, the triangles formedby joining their extremities are equiangular to each otlier I5» EUCLIDS ELEMENTS
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