30 INTRODUCTION. 



Scholium. Hence it may easily be shown, that multiple and 

 proportionate angles are subtended by multiple and proportionate 

 arcs. 



137. Theorem. If two chords of a given circle inter- 

 sect each other, the rectangles contained by the segments 

 of each are equal. 



^ — ^ Join AB and CD. Then ^AEBzrDEC (96), 



f /\\ and Z.BAE=DCE(133), both standing on BD, 



^L ^ — -Jd therefore tlie triangles AEB, CED, are similar, 



\\ / J and AE : CE: :EB : ED (121), therefore AE.ED 



^<4-^ =:CE.EB (123). 



138. Theorem. The rectangle, contained by the seg- 

 ments of a right line, intercepted by a circle and a given 

 point without it, is equal to the square of the tangent 

 drawn from that point. 



D Join AB, AC ; then ZABCzzCAD (134), and 



the angle at D is common, therefore the triangles 

 ABD, CAD, are similar, and BD ; AD: :AD ; CD 

 (121), whence BD.DC=:ADq (123). 



139. Theorem. In every triangle, the sides are as the 

 sines of their opposite angles, the radius being given. 



C Take AB=CD, and draw BE and CF per- 



I5^,.-^l\ pendicular to AD, then they are the sines of 



^ J ^ the angles A and D, to the radius AB or CD 



^' 1^ (130), and by similar triangles, AC:CF:: 



AB ; BE (121), or CD : BE. And the same may be shown of the 

 other sides and angles. 



140. Theorem. The sineofthe sura, or difference, of 

 any two arcs, is equal to the sum or difference of the sines 

 of the separate arcs, each being reduced in the ratio of 

 the radius to the cosine of the other arc. 



