A Collection of Formulae for the Area of a Plane Triangle

Annals of Mathematics

A Collection of Formulae for the Area of a Plane TriangleAuthor(s): Marcus BakerSource: Annals of Mathematics, Vol. 2, No. 1 (Sep., 1885), pp. 11-18Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1967393 .

Accessed: 26/05/2014 19:36

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals ofMathematics.

http://www.jstor.org

This content downloaded from 194.29.185.178 on Mon, 26 May 2014 19:36:03 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=annals

http://www.jstor.org/stable/1967393?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


BAKER. FORMULA FOR THlE AREA OF A PLANE TRIANGIE. II

we obtain as their internal product

a I b - (a, C + (a2e2) (19le2 - A2el) = (a631 + ({222) '1t'2

_ a1id, + (E2X2A

Similarly we find for the internal square

a I - a 12 + 2

[TO BE CONTINUED.]

A COLLECTION OF FORMULAE FOR THE AREA OF A PLANE TRIANGLE.

By MR. MARCUS BAKER, Washington, D. C.

[CONTINUED FROM VOL. I, PAGE I 38.]

GROUP II. PART I.

32. 4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 32. -lJ/87flc2 (a2 ? b2 -2m~c2) -(a2 b2)2

33. 21kc ( Ia2hC2 + I/ b2 /c2)

34. 20ab + bX( + -

35 31//4s7,2 - _ [9(12 (,;2 + m 2)]2 2 2 _- k4

where 2kb,2 - a2 (mb2 + mC2), etc.

36. lb a -hb - hi C/a2 - 2

2 (/'I 2 - /C2)

37. 1 (V4111nb2 - h1a2 + V/4M,2 -a )

38. 2ak,.

39. 21/ab1 b 1I _Ib h a - kb



12 BAKER. FORAIUIL.1" FOR THE AREA OF A PLANE TRIANGLE.

GROUP I1. PART 1.-COlltib/u1d.

40.* a / 12 + 1/ (1m1a2 /1a 2) (9,b2 -/12) [) ma2 /1,12

4I1. -iia1a -f- ma2 + a Vf 1)1 -22

+ [ +a- + I /12-a1/v12/2)

-4I (v'ja2. ? ka2 ? ak + li-a2 ? ka 2 -a,' where ka2 = - 11a2 etc

Ri/b/I. 42. a

ab 2- 2 --2 43. jta V/2 a2+2 - R/ (2mta )2)

44. h.a(1/'b 2 -/i2?/ mazka2)I

aba

45.-t 8R2(ajll4R - b + b 1/4R2- 2)

46. 2R2 [(f - 2X2) (g- 2X2) (I-g+ 212X2)

+ (g - 2X2) /(f-2X2) (If + 2X2)],

wheref = 1-12+ 21,g2 2111a 2+ ,

2 and x (= sin C) is to be found from

the equation

rV(f 2*2)(I g + 2X) + I/(g 2x2) (I -f + 2X*).

si Sara

2 (/ha+ra)

*If ,u and 0 are auxiliaries determined from the equations cos = ha and cos ha then Oa

h(,2 sin 2(. s-)

cosyu sin 20

[For this elegant expression I am indebted to my friend Mr. Charles H. Kumniell, of the U. S. Coast and Geodetic Survey.]

tF7R2/ha2 - A2 + V/ _2 = I r R'aki ) 2



BAKER. FORMIUL.E FOR THE AREA OF A PLANE TRIANGLE. I3

GROUP 11. PART II.

48. 2ab sin C

s in C

2sin B sin C la2 5 0. 1a2 . 2

sin A cotB + cot C

51. /~2 sinA -1_ 5 a 1 B-siC 2kla (cot B + cot C) - I/"2 (sin 2B + sin 2C)

52. sinb2). si B=( Si2 2B+ 2sin2A) cot A-cot b

53. sinsinBB g 1,117tan iAstanBstan 2

1 1tan tan 2-B tan C cos A (A - tan2 4-B)

2ma2 sin A sin B sin C 54. 2(sin2AX+ sin2 B-+-sin2C)-3 sil2 A

2 (ml2 m 22) 550 * 3 (cotA- cot o)

Vb22 [R2 2 A-

57. -R- sinA1 sinB A sC(- sin2 A ? 2sin2B + 2 Sil2 C)

4m( 2 + 3a2 5. 8(cotA + cot B + cot C) 4 (ma2 +

Jni2)-3C2 59 4(cot A + cot B-+ cot C)

56. 1/4bsinA2 [R i A 11b2+1,,2]

6o. 24a2 sin B sin C sin (B -i 2)sin (A s + 2 B)

6i. R2a sinA

62. 2a9( sin (C ? 2A)

63. R( siA 2 ( I + COS C)

64. Ra sinBsin C Ra (cost + cosBcosC)

65. Riadtan B sin B tan 2

66. R1i tan A sin B tan 2 Csin (C + 1 A)



I4 BAKER. FORMUL.-E FOR THE AREA OF A PLANE TRIANGLE.

GROUP II. PART Ii.-Coil/lsz1'1 d.

67. "a ('lb +

/lc) 4sin(C+ A)cOs A

68. l~a (/lb + /c) 4 COS 2

69. i,2(b+c)4 + Jtan'A

70. 2j~atbsin (B + -A) sin (A + 1B) sin C

- sin 'A + /3b sin >RB + d sin - C 71. j aI jc - (b!cC COS SA ? dcdacOS JB + 18a9b COS IC

72. la sin B (a COS B + i/b2-2_ a2 sin2 B) lb sin A (b COS A ? v/a2 -b2 sin2 A)

GROUP III. PART I.

73. sr ~(s-a)rra

74. Jr (v/"2ma + 211lb -M m+ 1V2mb2 + 2mc2 - Ina2 M1c/2lfl2 + 2ma2 mb2)

=ira (1,/2ma2 + 2nb2 - ,2 - 2mb2 + 2mc2 ma2 ? V/21 lc2 ? 2ma2- mb2)

a 711 _ M _M a _ _Mb__

75. + rrbrc _ rarbrc

ia fb ~ a h b h + 'la/lb/c ha kb/c

R R 76. 25(ahb+ b? ca) 2(s a) (/a/lb /bc + hcha)

77. a jRr ('la/b + hb'hc + /kcha) 1 IRra (ahb - , i)c +h h ic/a)

78. s s a

I + + I I +I +I

k, b hc h,, hb h,

I r

(h h)h a h a hk+he)(fb+ hr

ra

hI hb + / ) hb +he ) ja /ib he)



BAKER. FORMUL.i FOR THE AREA OF A PLANE TRIANGLE. I 5

GROUP III. PART II.

80. a(- sasin 'A

+l9bsin 2B + P3Csin

C)

2S(t3asin 2A + 9, sin 21-B + P, sin IC)

8 1 | 2r t? _ _cos (A_-B) + cos (B- C) + cos (C- A) + i] 8i.\/ rIIa~bI9[ ( -cos A + Cos -B + Cos C- IJ

4 2[A9 9 [cos (A -B) ( + os - C) + Cos (C - A) +

82. S2tan 'A tan 2B tan IC=-(s -a)2 tan JA cot JBcot IC 83. r2 cot 2'A cot jB cot jC= ra2 cot jA tan JB tan I C.

GROUP IV. PART I.

84. a rrra _ a rbr, ra -r rb + rc

rb + r, r - r 85. rra = rbrc a

86. (a + b) r;c = (a - b) ralb r + r ra-rb

rb-rc r + r,, 8 7. r rab C=rr

/ b + bc ra-c 88. rra\rb ^+r = rbr a \rb

89. rra4 -(ra- ) =1brC /4- (rb + I-) N ra - r l'b ~~+ rC

9o. shara (s - a) h~r ha+2ra haa-2r

91. r4 1 h1rbr ra I arbrc

a - 2r \ha + 2% a

92. r ra

akb + h + h

rbrc

/lan + h, +rl h.,8( z ,. +b C



i6 BAKER. FORNIUL.-E FOR THE ARE-\ OF A PLANTE TRIANGLE.

GROUP IV. PART II.

J

93. rr,, cot -A - rhr, tan A-4

GROUP V.

94, r2 cot AA + 2Rr sin A - a2 cot A-A - 2RrI sin A - r,2 cot AA - 2Rr,, sin A

'r cot I A - 21R; sin A. MISCELLANEOUS EXPRESSION'S FOR THE AREA OF A PLANE TRIANGLE.

If we designate the distances from the orthocentre to the sides of the triangle by ka, k,, k, and from the orthocentre to the vertices by kq', k,', k/, then

95. J = I (aka + bk, + ckc) 96.* 4 = - (aka/1 t Mkb' + ckc)

97.* 1 81? (a2ka,' cosec A + b2kb' cosec B + c~k,' cosec C).

If we designate the distances between the centres of the escribed circles by d(,Z db, d, then

98.* J = j/2ddbdr sin A sin B sin C

d. dadif4r 99- J I 6R2

IOO. J 'djfR-sin A- sin A- sin IC

IOI.* j rarb4\( + 2--

If x.Yj, xbyb, xy. designate the rectangular co-ordinates of a triangle in a plane, then

I02. J = I -Ya (Xc. -Vb) + Yb (-Va - X) + Yc (Xb Xa)]

and, in polar co-ordinates,

I03. J = I [r(,rb sin (6 - 6') + i-br, sin (0"' - 6") + rera sin (6' - 6"')].

If the equations of the sides of a triangle are

aax+ baY +ca O; abx + by + c= o; ax + bey + co,

then

I04. - = [[ aa ((bc - becb) + ab (bcCa - baCc) + ac (b(,cb- bbca)]21

(a,,bb - baab) (abbC - bbaC) (acba - bcaa)

* Tidsskriftfor Mathematik. 8?. Copenhagen, i884, 5th Series, Vol. I, No. 2, pp. 78-79.

Mathesis. 8?. Gand, i884, November, Vol. IV, p. 232.



BAKER. FORMULAE FOR THE AREA OF A PLANE TRIANGLE. 17

If the co-ordinates of the vertices of a triangle are x.,Ya, Za; Xb,,Yb, z,; x,, y, zC, and

a aX + Ya2 + Z12% A XbXC + YbYc + ZbZe,

z Xb2 + Yb2 + Zb, / Xc"a + YcYa + Zcra,

zC2 ~y2 +-Zf2; v x.;xb+ yl, +zazb,; then

0 I I I

I05.* 4J2 = I a

~I , i 2r

or, without determinants,

J2 2a2b2 + 2b2c2 + 2cla2 - a4 b4 - c4

where a2 (Xb X-)2 + (Yb - y")2 + (Z_ Z )2

b2 - (x. Xr)2 + (YC _ ya)2 + (Z Z. )2

(X. - Fb) + (Ya _ yb)2 + (Z. -z )2

and if a, P, r. 2, p and v, have the values set down above, then

a2 = + r - 2A,

b= r + a - 2/1, C2 a + 2- P,

and J2 2 (a + P 2)(P + r 2A) (aP+ P 2)2 + 2( + -2A)( + a 2) - (+ r- 22)2

+ 2 (r + a- 2)(rx + P 2v) -(r + a - 2/1)2,

which reduces to

=2 4(2 +u p+ )2 - 8 (a + 2P + r7) + 4 (a0 + Pr + na).

IO6. J8R[(-s - Da) + (s-b) Db2+( c)DA2-sD2] where D, Da, Db,

D, are the distances between the circumcentre and the in- and e-centres.

07. J 4K, where K is the area of the triangle formed by joining the feet of

the medians.

*Giornale di Matenzatiche, Naples, i884, July, Vol. XXII, pp. 207-8.



i8 PROBLEM AND SOLUTION.

Io8. J= L where L is the area of the edal triangle or tri- 2 co A co B co whrC'ste rao h pedltinl rti

angle formed by joining the feet of the perpendiculars.

I09. J - M(a + b) (b + c) (c + a), where M is the area of the triangle formed iabc

by joining the feet of the internal bisectors.

2R IIO. a = r N, where N is the area of the triangle formed by joining the

points of tangency of the in-circle.

CONCLUDING NOTE.-Owing to some changes and additions in the foregoing article made since its original preparation and while in press, the statement on page I35, Vol. I, that "the total number of formulae ' * * in this collection is ninety-three," etc., is no longer true.

The number of formulae in the collection is shown below where the second column shows the number that result from counting as distinct those formula arising from permutation.

Group I, 3I 3I Group II, 4I I23

Group III, II 44 Group IV, IO 6o Group V, I I2

Miscellaneous Group, i6 i8

Total, II0 288

[In the original manuscript all the forms arising from permuting the letters were given in full. To save space in printing, those forms which arose from mere cyclical permutation were omitted.-O. S.]

PROBLEM AND SOLUTION.

By DR. J. E. HENDRICKS, Des Moines, Iowa.

PROBLEM -To inscribe a rectangle whose sides are c,x in another rectangle whose sides are a,b, and express x in functions of a, b, and c.



A Collection of Formulae for the Area of a Plane Triangle

Documents

Transcript of A Collection of Formulae for the Area of a Plane Triangle