Annals of Mathematics

A Collection of Formulae for the Area of a Plane TriangleAuthor(s): Marcus BakerSource: Annals of Mathematics, Vol. 1, No. 6 (Jan., 1885), pp. 134-138Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1967384 .

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134 BAKER. FORMULiE FOR THE AREA OF A PLANE TRIANGLE

troids and axoids (Liouvi/li's Journal, XVI, 9-129 ; 286-336). The applica- tion of these methods to the study of Mechanisms was first made systeirnatically by Willis (1841); his processes were improved by Rankine (Machzinery and Mil//work, i87i); and the whole subject was reformed and freshly stated by Reu- leaux (Kinernatik, 1874), and has since been still further developed by Grashot and others. Peculiar interest attaches to the "tram-motion" from its furnishing to Leonardo da Vinci his famous discovery of the elliptic chuck for turning ovals on the lathe (Chasles, Apcri 5 3I). -W. ML7. T1]

A COLLECTION OF FORMULzE FOR THE AREA OF A PLANE TRIANGLE.*

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

In April, i883, Mr. James Main, formerly of the U. S. Coast and Geodetic Survey, published in the Mfathematical Malezgazine a collection of forty-six (46) ex- pressions for the area of a plane triangle, prefacing it with the remark that this collection "may be regarded as a matter of curiosity," and that about one-half of the formulae are well known.

In the following August M. Ed. Lucas reprinted this collection in Matliesis in a classified form, separating the formulae into five groups and adding one for- mula not contained in Mr. Main's list. The collection has also been reprinted in the third number of the Tidsskrijt for A'Jat/emnatik, i883. Some two or three additional formulae have since been printed in various mathematical publications.

The terms in which the area is expressed in Mr. Main's collection are angles, sides, perpendiculars, and radii of inscribed, escribed, and circumscribed circles. No formulae are given involving medians or bisectors. In numbering Mr. Main has not counted those formulae as distinct which arise from merely permuting the letters, nor has he in every case given all the forms possible to be obtained by permuting the letters, though he has Generally done so. As numbered, then, he counts forty-six fortnulae, but if every form be counted as a distinct one the total number is ninety-four.

M. Lucas, by making all possible permutations and adding one new form, makes the number 139, to which sonme two or three have been added since.

As the matter has proved of interest, the following collection has been made, which is a still further extension; the additional formulae being chiefly due to introducing the medians and bisectors, not used in the former collections. In this collection Mr. Main's mode of numbering has been followed and formulae

*Read before the Mathematical section of the Philosophical Society of Washington, January 7, I885.

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BAKER. FORMUL.,E FOR THE AREA OF A PLANE TRIANGLE. 1 3 5

derived by permutation are not enumerated as distinct formula. Moreover, for- rnule expressed in the same terms, but in different form, are also considered as but one. For example, in the former lists we find

J = 2R2 sin A sin B sin C,

J - I 2 (sin 2A + sin 2B + sin 2C),

and I - 2R2 [sin3 A cos (B - C) + sin3 B cos (C -A) + sin3 C cos (A - B)]

given as three distinct formul Xwhereas they are here counted as one, and the principle involved herein is employed throughout.

The total number of formulae for the area of a plane triangle in this collec- tion is ninety-three, not counting those arising from permutation. If these be counted as distinct the total number is two hundred and sixty-nine.

Owing to pressure of other duties and consequent lack of time some of the groups in this collection are not so fully worked up as had been planned.

It may be noted that a number of curious and interesting theorems may be obtained by equating different expressions for the area and reducing. In this collection we have classified, for convenience, all expressions for the area into five groups and each group into two parts.

Group I contains formulk which M. Lucas has called ziraquc, i. e. formulae which do not admit of other similar formulas by merely permuting the letters. In such formula all the sides, all the perpendiculars, all the medians, etc. must enter if one enters.

Group II contains formula which admit of tewo similar expressions by per- mutation giving three of a kind.

Group III contains formule which admit of three similar expressions by per- mutation giving four of a kind.

Group IV contains formulae which admit of five similar expressions by per- mutation giving six of a kind.

Group V contains formulae which admit of le;t?7e similar expressions by per- mutation giving twelve of a kind.

Each group is divided into tzvo parts, the first contailing and the second siot containing trigonometrical functions. In addition to the foregoing a group of miscellaneous expressions, not falling within the classification used, has been added, and called the Miscellaneous Group.

The notation used is as follows:

J - the area of the triangle; A, B, C- the angles;

a, b, c = the sides opposite A, B, and C respectively s = semi perimeter 2- (a + b + c);

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I36 BAKER. FOR.MUL.-E FOR THE AREA OF A PLANE TRIANGLE.

A, r, i1, ri-,, i, = the radii of circumscribed, inscribed, escribed circles respectively; ha,, lb, hc - the perpendiculars from A, b', and C respectively;

//I,,, in,,, In, /the medians from A, B, and C respectively; ,<, ,ji, ic-the bisectors (internal) of the angles A, /3, and C respectively; and

-2 (In, + nib + Ille).

GROUP I. PART I.

I. V S (s - az) (s-b) (s-c) V/2a2b2 V+ 2b2c' + 2 c a- at -eb-c4

2V 43 1 ' a (d = StZ) (d 1tlbj (a - t4 4)

31 2tl tJl,, + 2lil6lt + 2iilcli2a - 1iia - ilI 1-il.4

3. I 3

( t / /Zb AZ + ha 1kb /Ic L/'Z, b/ l / )

Ag2Ab2/1,2_

V (/1 a/lb-V +//Ic +-/I,/Ia) (/l1Ia/b t/I b/I-Vc/+' a ) (/11al/b /b/c -+V/la) (/11a/l+ -Ib14c/ -c/la)

4. rrarbrc

abc 5 4R

6. -- V"(21ina2 + 21iib2- iljc2) (2i1 +22 ? 2iic2 i n2) (211ZC2 + 211Z2 iib2)

27 2 -21 111b/b/

7. V _R/l __l/l

8 r4(ia + rb)(rb + ri)(re + ra) 2\ R

9. 2 fl3abckt/t~z 1. -- =bc/la/lb/Ic

I0. I .bc _3 re (+1, 1tti/,) (r, -V is) (is-, + is) -* a 2 -V 62 + C2 U r(, +ri,)+"-Vir aIbc- ----

I _ _ _ _ _ _ _ _ _ _ _ _2 --- 12--/ 1 "

3/IZ /IZb/I c/3 (lila V iili2 + 111 c) (k/1/1,2)

+ /162/Zc2 t- /12/-2)

abc s _

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BAKER. FORMULA FOR THE AREA OF A PLANE TRIANGLE. 137

GROUP I. PART i.-Continzued.

13. ( RI 2 + 2b2 + M,,2) (/k2/i2 + lib/I2+ F /2/lt2) J 6

dKtfcz a + b b + c c t a I __II__ '4.- 4- ?b (7jb S 2C 2a 2b a b + a + b + c I I

2 ? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~112 2 2]2 2 2, I5-3i~<Xi}2{ _V2Ma +2Mgb Mc 2//b 2H~ -- 2MZC 2} M2 + 2Z 2 ltlb

1/ 2 M,2 + 2 IU-m2 M,2 ? 1 /2M62 +? 2111, - ;nQ ? 1/2111,2 + 2lacL2- I/hz]

L?L ha + L?L + +

I7. dt-nidct l+ + I I r;

L r, rb Frb r. r. r.

I8. <ryc3ji3c [(a + b + ) I]

I9 l.jaIbc (a + b) (b + c) (c + a)

^0 ARr ( +i ) (a + b + +

hail b/i2 2 I . r-8al-'(r(, + rb + r. r) .

2. 8abc

GROUP I. PART II.

22. 2R2Sin A sin B sin C

- R2 (sin 2A +- sin 2B + sin 2 C) 2R2 [sin3 A cos (B - C) + sin3 B cos (C- A) + sin3C cos (A-B)j

23. 2R(a cos A + b cos B + c cos C) R [a COS Ccos A + b COS A COS B + c COS B COS C]

24. _(a2 cot A+ b2 cot B + c2 cot C) = 1 cotA 2 +cot B cot C

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I38 HALSTED. DEMONSTRATION OF DESCARTES'S THEOREM AND EULER'S THEOREM.

GROUP I. PART II.-CGoftilucd.

Ma * m2 + o112 + 11, 2- ma V -,

-4-- ;j

Tcot A + cot B + cot C

26. V2Ra3aj3cji3 [I ? COs (A- B) + cos (B- C) + cos (C- A)]

- 2/R31j,3,,i sin (A? - 2B) sin (B + -C) Sin (C + A)

27. RA-/I~/biI. siA sin B sin C

28. -R sin A sin B sin C -b- +

29 ~ ~~~~ + /4v + A 29. _ __ _alb+/ _.

2(cos 'A cos 1B Cos {C~ 2 (_32 + C?_- 2 + CS2 )

j@, sin (C + 'A) + 3, sin(A + ? B) + 3, sin (B + i-C) 30.2 L A ? cos .jB cos C) 2 2- --F +--- -__ __

I jaI3bjscl/1a I -/2) (o;j72 - /it2) (;;z2 _/g2)

s(a -b) (b-c) (c - a)

[TO BE CONTINUED].

*TidsskriftforMathemzatik, 8? Copenhagen, i883, fifth series, first year, No. 4, p. 136.

DEMONSTRATION OF DESCARTES'S THEOREM AND EULER'S THEOREM.

By PROF. G. B. HALSTED, Austin, Texas.

DESCARTES S THEOREM.

Cutting by diagonals the faces not triangles into triangles, the whole surface of any polyhedron contains a number of triangular faces four less than double the number of summits.

Prtoof.

For, joining all the summits by a single closed broken line, this cuts the surface into two skew polygons, each of which contains S - 2 triangles, where S is the number of summits.

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