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If HG and FI b: drawn through any point E of the Diigonal CB parallel 10 AC and CD, the Parallelogram ACDB is divided into four Parallelograms, two of which are about the Diagonal, and the other I wo are their Coinplements, which are thus Thewn to be qual, a ABC = to a CDB and the as HB E and EFC are = to as IBE and ECG (by 8 ) If from the Equal AS AB and CDB vou subftra& the Equal As HBE, EFC and IBE, CEG there will remain the Parall-logram AHEF = to the Parallelogram EGID (by 2d Axiom.)
T HE OR EM XII.
Fig.52. In every rectangle Triangle A BC, the Square of the side AC, which is opposite to the right Angle, is equal to the Squares of the other two fides (A B, CB)
Demon. D:aw IC.BF and BE,Parallel to AF. I then you add the common < BAC to lhe right Angles IAB, FAC, and therefore egial, the wholes TAC, FAB, will be qual, but the as IAC, FAB have the two fides which contains those Angles equal (by Def.15) to wii IA = BA and CA = FA. IAC
A FAB (by Prop. 1) but a TAC = 4 thtag are ILBA and ABF= Parallelo
gram AFZ the (by Prop.10) therefore Square LIBA =the Parallelogram AFZE. I might te shewn with the same Ease that the Square BXCH =the Parallelograin CZER. Q. E D.
Fig.53. An <(BCA) at the Center is double to the < (AFB) at the Circuinference when the same Arc (AB) is Bife to both Angles. This Prop. hath three Cafes, The first is when the fide (CA) coincides with the side (AF). For then CF = CB, because both ara drawn from the Center to the Circunference of the fame Circle therefore in A CFB CBF =< CFB (per Prop. 2) but <BCA== CBF | < CFB (per Schol. 7 Prap.) : 5 ACB is dou' le the <CFB which may the first In the second Case CA and CB iall with. out AF and BF. Then <XCA is double < A FX, and < XCB is doubl the <XFB (by the fifi Cafe) Thesetore the widole <ACB is donble the whole < AEB. In the third Case RK cuts CA, and the < AKB is wholly without the < ACB Draw KCL then <ACL is double < AKL (by the filt Cafe) and if <LCB and its double <LKB be taken away there remains <A B doub.e FAKB. Q. E. D.
All similar Triangles have their fides about their egual Angles proportioral
. For if they were inscribed in Circles, their Şides would be Chords of finilar Arcs.
F8.54. If in any Triangle a Line be drawn parallel to the. Bate, that Line will cut the Legs proportionaly. In the Triangle ABC let the Line DE, be pasallel to BC: I say that D, is on AD as AB to A and AB: BC:: AD: DE. Also DE: BC:AD: AB:or AD: AB :: AE: AC. For As ABC, and -DE are similar because <D = < B and <E= <G (by the 6th) and < A is common to both .. their Sides about their qual Angles are proportional (by the luft) Q. E. D.
Of Right Lines applied to a Circle.
1. Every Circle is supposed to be divid. ed into 360 D g. and each Dag. into 60 parts, called Minutes, and each Minute into 60 parts, called Seconds, &c. Any Portion of the Circunference whereof is an Arch,
and is Measured by the Number of Degrees it contains.
2. Fig. 55. A Chord is a Riglit Line joyning the Extremities of an Arch, as AC, is the chord of the Arches ABC, ADC.
3 A Tangent of an Arch is a Righ:line drawn Perpendicular to the end of the Radius or Semi D'ameter, palling through one end of the Arch, and its length is limited by a Rightline drawn for the renter through the other End of the Arch, which is called the Secant; thus BM is the Tan. gent, and FM the Secant of the Arches AB and AD.
A Right Sine is a Right-line" drawn from one End of an Arch, Perpen.' dicular to that Diaineter passing through the other End, or is half the (hord of the double Arch; AE is the Right Sine of the Arches AB, and AD. And here is evident, that the Sine of 90 Dug. which is equal to the Radius, or Semi-diameter of that circle, is the greatest of all Sines, the Sine of au Arch greater then a Quadrant, being less than the Radius.
5. A Versed Sine is the Segment of the Diameter intercepted
between the Arch, and the Right Sine, EB!is the Versed Sine of the Arch AB, and ED of the Arch AD.
The difference of an Arch from a Quadrant,, whether it be greater or less,
is call'd its Complement, GA is the Complement of the Arches AB, AD; HA is the Sine of the Complement, or Cofine, GI the Tangent of that Complemen', or Cc-Tangent, Fl the S. cant of that Complement, or Cc-secant.
Is the Menfuration of ihe sides and Argles of plain Triangles. A plain Triangle has six parts, viz. "Three Sides and three Angles, whereof any three being given, eso cept the three Angles, the other may be tound by Trigonometrical Calculation.
in right Angled Triangles, there are ftven Cases, all performed by the following .cxioms,
In any Right Angle Triangle, if either of the Legs be supposed to be the Radius of a Circle, the other Log will be the Tangent of the opposite Angle or of the Angle at the Center, and the Hypothenuse will be the Secant of that Angle : But if yon in agine the Hypothenuse to be the Radius of a Circle, then each Leg will be the Sine of its opposite Angle, or of the Angle at the Center, as is plain fron. Fig. 56.57:58:59.