Elements of geometry and trigonometry . PROPOSITION VII. THEOREM. If from the vertices of the three angles of a spherical triangle, aspoles, three a?cs be described forming a second triangle, thevertices of the angles of this second triangle, will be respectivelypoles of the sides of the first. From the vertices A, B. C,as poles, let the arcs EF, FD,ED, be described, forming onthe surface of the sphere, thetriangle DFE ; then will thepoints D, E, and F, be respec-tively poles of the sides BC,ACAB. For, the point A being thepole of the arc EF, the dis-tance AE is a quadrant ; thepoint C being t
Elements of geometry and trigonometry . PROPOSITION VII. THEOREM. If from the vertices of the three angles of a spherical triangle, aspoles, three a?cs be described forming a second triangle, thevertices of the angles of this second triangle, will be respectivelypoles of the sides of the first. From the vertices A, B. C,as poles, let the arcs EF, FD,ED, be described, forming onthe surface of the sphere, thetriangle DFE ; then will thepoints D, E, and F, be respec-tively poles of the sides BC,ACAB. For, the point A being thepole of the arc EF, the dis-tance AE is a quadrant ; thepoint C being the pole of the arc DE, the distance CE is likewise a quadrant : hence the point E is removed the length of aquadrant from each of the points A and C ; hence, it is thepole of the arc AC (Prop. V. Cor. 3.). It might be shown, bythe same method, that D is the pole of the arc BC, and F thatof the arc Cor. Hence the triangle ABC may be described by meansof DEF, as DEF is described by means of ABC. Triangles80 described are called polar triangles, or supplemental tri-angles. BOOK IX. 193 PROPOSITION VIII. THEOREM. The same supposition continuing as in the last Proposition, eachangle in one of the triangles, will be measured by a semicir-aimference, minus the side lying opposite to it in the othertriaji[rle.
Size: 1729px × 1445px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
