. Elements of plane and spherical trigonometry . the four triangles,taken in the aforesaid order, we write r, r,, r,„ r,,/. whilst, for theladii of the circumscribed circles, we put R, R„ R^^, R^^,, unaccented letters referring to the circles of the fundamental tri-angles. These triangles possess many beautiful properties when consideredm their mutual association, which render them worthy of greater atten-tion than has yet been bestowed upon them. Indeed, till very recentlytheir existence has scarcely been alluded to by writers on sphericalsubjects, and even to the present day
. Elements of plane and spherical trigonometry . the four triangles,taken in the aforesaid order, we write r, r,, r,„ r,,/. whilst, for theladii of the circumscribed circles, we put R, R„ R^^, R^^,, unaccented letters referring to the circles of the fundamental tri-angles. These triangles possess many beautiful properties when consideredm their mutual association, which render them worthy of greater atten-tion than has yet been bestowed upon them. Indeed, till very recentlytheir existence has scarcely been alluded to by writers on sphericalsubjects, and even to the present day, not more than three of their pro-perties have, we believe, been published. 2. Let O be the centre of the circle in-scribed in the fundamental triangle, and G,H, K, its points of contact with the sides, joinAO, BO, CO, and draw the radii to the pointsof contact. Then the tangents from A to thecircle are equal; that is, AK = AH; in likemanner BK == BG, and CG = = AH = aPut ^ BK = BG = /5LCG = CH = y From which Whence ? Lc = a -f + /? + ^ = y = a-\-b-\- c 2— a-\-b-\- c 2a — b -\- c 3 a-\-b — c (1). Again, in the right-angled triangle BOG, we havetan. OG = sin. BG tan. OBG, that is, tan. r = sin. 0 tan. § B; or by (1)just given and (3), upon page 49, applied to B, we have tan. r sin. 5 — * sm — a sin. s 5 — -b _V ^sin. 5 sin. s — a sin. 5 - -b sin. 5 — c . (2). sm. s Again, in the supplemental triangle BAC, denoting the quantitiesBB, CH, and AK, by /?/, y^, a^ we shall have «. = /?/ + y.*/ = «/ + y, and hence, as before, s, —a,^(ij-{- y, — a,-\-b,-\-c, a -\-TT—b-\-7r—^ —«-f*-j-c 3— = 2 = 2 = T 5 — «, 128 SPHERICAL GEOMETRY. -a,^b,2 a — ir- — a-x-tr— J -f rr — c a-\-b^c- 2 s,—b,=P, b + c - 2 «/ -b-{-n—c a-\-b V r 2 -\-b, — c^ a-{-iT 2 - 2 a^ S/ —C/ = y/ — b— TT — c a — = 5 — 6. = s> Also B/ = TT — B, and tan. J B, = cot. I B. Hence, in the right,angled triangle BOG, we have tan. OG^ sin. BG tan.
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Keywords: ., bookcentury1800, booksubjectnavigation, booksubjecttrigonometry
