Elements of geometry and trigonometry . PROPOSITION XX. THEOREM. The angle formed by two secants, is measured by half the diffe-rence of the arcs included between its sides. Let AB, AC, be tw^o secants : thenwill the angle BAC be measured byhalf the difference of the arcs BECand DF. Draw DE parallel to AC : then willthe arc EC be equal to DF, and theangle BDE equal to the angle BDE is measured by half the arcBE ; hence, BAC is also measured byhalf the arc BE ; that is, by half thedifference of BEC and EC, or half thedifference of BEC and PROPOSITION XXI. THEOREM. 77/c angle formed
Elements of geometry and trigonometry . PROPOSITION XX. THEOREM. The angle formed by two secants, is measured by half the diffe-rence of the arcs included between its sides. Let AB, AC, be tw^o secants : thenwill the angle BAC be measured byhalf the difference of the arcs BECand DF. Draw DE parallel to AC : then willthe arc EC be equal to DF, and theangle BDE equal to the angle BDE is measured by half the arcBE ; hence, BAC is also measured byhalf the arc BE ; that is, by half thedifference of BEC and EC, or half thedifference of BEC and PROPOSITION XXI. THEOREM. 77/c angle formed by a tangent and a chord, is measured by halfof the arc included between its sides. BOOK III. 57 Let BE be the tangent, and AC the cliord, From A, the point of contact, draw thediameter AD. The angle BAD is a riglitangle (Prop. IX.), and is measm-ed byhalf the seinicircumlerence AMD; theangle DAC is measured bv the half ofDC: hence, BAD + DAC,* or BAC, ismeasured by the half of AMD plus thehalf of DC, or by half the whole arcAMDC. It might be shown, by taking the difference betweengles DAE, DAC, that the angle CAE is measured by 1arc AC, included between its sides.
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
