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Let the preceding figure and construction be res tained, and let AG and CG be drawn. The triangles AHD and GHC (being right-angled at D and C) and having 'HAD = HGC (by 11. 3.) are equiangular; and fo AD:GC:: AH : HG:: AE: AG (by 19. 4.); whence, alternately, AD: AE::GC: AG, and AD’: AE° :: GC2 (GFX HG): AGʻ (GE XHG):: GF :GE; therefore, by division, &c. AE-AD': AE :: EF: GE:: EFXAE: GEX AE; whence, again, by alternation, &c. EFXAE: AE — AD? (AE+AD X AE-AD):: GE XAE: AE :: GE: AE:: radius : tang. AGE (by Theor. 2.); from which the truth of the proposition is manifeft.

PROP. XV.

If the relation of three right-lines a, b and x, be such, that ax--*=bʼ; then it will be, as į a:b : : radius to the fine of an angle; and, as radius, to the tangent (or co-tangent) of balf this angle, so

is b: X.

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Make AB equal to a, upon which let a semi

circle ADB be described;

B also let CD, equal to by ΑΙ

be perpendicular to AB, and meet the periphery in D (for it cannot exceed

the radius of the circle when the proposition is possible): moreover, let AD, BD, and the radius OD, be drawn. Because ACXCB=CD=64 (by Cor. to 19. 4.) it is plain tbat AC X2 AC, or BC x2-BC is also =b2; and, therefore, xx@--* being =b, it is manifest that x may be equal, either, to AC, or to BC. Now

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(by Theor. 1.) OD (1a): CD (6):: radius : sine DOC; whose hali is equal to A, or BDC (by 10. 3.) But, as radius : tang. BDC :: DC (6) : BC; or, as radius : co-tang. BDC (tang. CDA) :: DC (b): AC. Q: E. D.

PROP. XVI.

If the relation of three lines, a, b and x, be such, that x' + ax =b; then it will be; as į aib:: radius to the tangent of an angle; and as radius is to the tangent, or co-tangent, of half this angle (according as the sign of ax is positive or negative) ::b:x.

D

Make AB=b, and AC, perpendicular to AB, equal

E to à ; about the latter of which, as a diameter, let a circle be described ; and,

A throo, the center thereof, let BD be drawn, meeting the periphery in E and D; also let A, E and C, E be joined, and draw BF parallel to AC, meeting AE, produced, in F. Then, fince (by 22. 3.) BEXBD (=BE X BE +a = BD BD -- a) =AB’ (b). and x x*+ a=b, by supposition, it is manifest, that BE will be =x, when *** +=b; and BD-x, when * Xx-a-o.

Furthermore, because the angle F = OAE (by 7,1.) =OEA (by 12. 1.) = BEF (by 3. 1.) it is evident that BF = BE (by 18. 1.) and that the angles BAF and C (being the complements of the equal angles F and OAE) are likewise equal.

Now (by Theor. 2.) AO({a); AB (6):: radius : tang. AOB; whose half is equal to C, or BAF (by 14. 3.) But, as radius : tang. BAF:: AB (6) F 2

: BF

.

: BF (BE), the value of x in the first case, wherë to2 + ax = b. Again, radius : co-tang. BAF (tang. F):: BF (BE): AB (by Theor. 2.); and BE CAB :: AB : BD (by Cor. to 22. 3. and 10. 4.); whence, by equality, radius : co-tang. BAF:: AB (6):BD; which is the value of x in the second case; where x — ax = b. Q. E. D.

PROP. XVII.

In any plane triangle ABC, it will be, as the linė CE biseeting the vertical engle, is to the base AB, jo is the secont of half the vertical angle A B, to the tangent of an angle; and, as the tangent of half this angle is to radius, so is the fine of half the vertical angle, to the fine of either angle, which the bisežting line makes with the base.

Let ACBD bë a circle de. fcribed about the triangle; and let CE be produced to meet the periphery thereof

in D; moreover, ler AD »FE A

B and BD be drawn, and like

wise DF, perpendicular to the base AB; which will,

also, bisect it, because (BCD being = ACD) the fubtenses BD and AD are equal (by 10. 3.) Moreover, since the angle DBE ACD (by 12. 3.) = DCB, the triangles DEB and DBC (having D common) are equiangular, and therefore DEX DC = DB: (by 24. 3.) or, which is the same, DE + DEX CE = DB:. Therefore (by Prop. 16.) CE : DB : : sadius : tangent of an angle (which we will call Q); and, as radius : tang. Q::DB : DE.

But DB : BF (AB) : : fecant FBD (BCE): radius; therefore, by compounding this, with the

first proportion, we have, įCEXDB : JAB X DB : : radius fecant BCE: radius x tang. Q (by 10. 4.) and consequently CE : AB : : fecant BCE : tang. Q (by 7. 4.) Àgain, DE: DB :; fine DBE (BCE, : fine of DEB (or of CEB); whence, by equality, tang. {Q : radius : : fine BCE : sine CEB. 2. E, D,

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PROP. XVIII. In any plane triangle ABC, it will be, as the perpendicular is to the sum of the two sides, yo-is the tangent of half the angle at the vertex, to the tangent of an angle; and, as radius is to the tangent of balf this angle, fo is the sum of the two sides, to the base of the triangle. Let DP, perpendicular

P to AB, be the diameter of a circle described about the triangle ; let CF be perpendicular to DP, and DG to AC, and let DA,

E DC and DB be drawn, and I A

B also FI, parallel to BD, meeting BA, produced, in I.

It is manifest, from Prop. 13. that CG is equal to half the fam of the sides AC and BC: it also appears, (from 7. 1. and Cor. to 12.3.) that the triangles DCG, ADE, BDE, and IFE, are all equiangular. Therefore it will be, CG2: BE':; DC' * (DF XDP): BD’ (DEX DP)::DF (DE+EF): DE :: BE + EI:EB:: BE + EIXBE:EB’; and, consequently, CG? (=BE + EI X BE) =BE' + EI X BE. But, by Prop. 16. it will be, įEI: CG : : radius : tangent of an angle (which we will çall Q); and as radius : tang, Q; : CG : BE

(: :

F 3

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(:: 2CG, or AC + BC, to AB). But (by Theor. 2.) EF: EI :: tang. I. (ACD): radius; therefore, by compounding this proportion with the laft but one, we shall have, El X EF:EI XCG:: tang. ACD x radius : tang. Q x radius (by 11. 4.) and confequently EF : 2CG (AC + BC) :: tang. ACD : tang. Q: Whence the truth of the proposition is manifeft.

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PROP. XIX. In any plane triangle ABC, it will be, as the perpendicular is to the difference of the two fides; fois the co-tangent of balf the vertical angle, to the tangent of an angle; and, as radiks is to the co-tangent of half this angle, fo is the difference of the sides to the base of the triangle. P

Let DP, DG, CF, &c. be as in the preceding proposition; also lec PA and PC be drawn, and Fl, pa rallel to PA, meeting AB

in I. А?

The right-angled triangles ADG and DPC, having DAG = DPC (by

Cor. to 12. 3.) are similar; and therefore, AG' : PC::: AD : DP2 : : AE ; AP (by Cor. to u. 4.); whence, alternately, AG' : AE : : PC° (PF X PD): AP (PE * PD):: PF:PE:: AI: AE::AI X AE: AEX (by 7.4.); and consequently, AGʻ = AIXAE= AE'-EI AE. Therefore, by Prop. 16, IEI:AG :: radius : tangent of an angle (0); and as radius : co-tang. Q::AG:AE. But ( by Theor. 2.) EF : EI : ; co-tang. EFI (ACD): radius ; which proportion being compounded with the last

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