Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 35-43 This paper is available online al hup://stdb.hnuc cdu.vn
S O M E IDENTITIES O F A T R I A N G L E D a m Van Nhi^ a n d L u u B a T h a n g ^
^School for Gifted Students, Hanoi National University of Education
^Faculry of Mathematics. Hanoi Nalional University of Education Abstract. The aim of this paper is to present some new equalities via the building of cubic polynomials. For instance, we introduce lengths of sides, altitudes and radii of excurles as the roots of cubic polynomials. Then, we use Vieta's formula and resultant and known triangle relations to build some interesting equalities and inequalities
Keywords: Geometry, triange, resultant, elimination, equality.
1. Introduction
Finding the equaUties related to the radii, altitudes and area of a triangle is always a classical and interesting problem of elementary geometry.
In the book Recent Advances in Geometric Inequalities [3], Mitrinovic et al presented the idea that the lengths of sides, altitudes and radii are roots of cubic polynomials, and then they collected and cited many interesting equalities and inequalities related to them. As far as we know, most of them are built taking the view of elementary algebra. In the view of modern algebra, which is taken in this paper, we propose and present the use of transformations of rational functions and resultants to build nice equatities and inequalities in geometry. Notice that these equalities and inequalities seem to be difficult if they are built from geometric property.
2. Main results
Suppose given a triangle AABC with the lengths of sides respectively being a, b, c.
Denoted the radius of the circumcircle as R, the radius of the incircle as r, the area of AABC as S, semi perimeter as p, the radii of the excircles as r j , r2, r^ and the altitudes of sides a, b, and c respectively as ha, hb and he- Then, we present and prove some results which are known and stated in [3, Chapter 1].
Received September 10, 2012. Accepted Octobers, 2012.
2000 Mathematics Subject Classification: 26D05,26DI5.51M16.
Contact Luu Ba Thang, e-mail address: [email protected]
Theorem 2.1. IS] With the above denoted, the following holds
(i) II. h. care three roots oJ the i nhic pohnomial .r' - 2px' 4- (ji' + r'' + 4Rr)x-4Rrp.
S'+4rir' + r- 2 23'
„.,,,, , ':. , :. air' ::^rT, •' +-
n '
(Hi) r,. ro.;;, are three iimf. of the cubic polxiioniial r - (iH + r)r' + [i' r - p'r.
I r „ 2 t a n ( . 4 / 2 ) Proof, (i) Because, tun - - and a = 21! sm A, we have a = 2H——-—, .
2 /< - rj 14- tan'(>l/2) Thus, we obtain
„ = tlir '''-"' => li' - 2pa' 4- (p' + r^+ \Iii )a - iRrp = 0.
( • - I ( / ) ii)-
Similarly, we have b' - 2iili' + (;r -I- r' + l / ? / ) i - !//(;) = 0 and c" - 2pc^ 4- (;<- +r^ + 4/?i)r - 4Rrp = 0. Therefore, ii.b&nd c a r e three roots of the (.quation
x^ - 2/u- 4- [p' 4- r ' 4- ARr)i - ARrp = 0 (2.1) (ii) Because a, b and i are the roots of the equation (2.1) and S = pr, we deduce that
s'-
2 5 2S 2 5 25^ T^ "*" ^^^ "^ ^'
— , -— and — are the roots of the equation 2.S'- y+ ~ 1/^ - Ry^ = 0.
a b c r 2 ' Hence, ha, hi,, h^ are three roots of the equation
3 S'' + 4Rr-> + r' 2 5 - 2.S'- „
' - 2Rr' '- + l ^ - ' ' l f = ° - <2-2' .1 r 2 *™ 7
(iii) Because tan ^ = — and a = 2/i' =—, . we have a = 4j?ri ., „. Otherwise 2 " 1 - F t „ i r l i '•' + P '
2
S = ri(p - a) = rp, we deduce Uiat a = — = ~ . Therefore,
4 « ^ ' ^ = ~f^ ^ ( n - r ) ( r ? + p ^ ) = 4 f t r ? ,
that means ri is a root of the equation x^ - (4fl 4- r j i ^ 4- p ^ i - p^r = 0. Similarly, r2 and rs are also roots of the equation
•':^-(4:R + r)x^ + p''x-p'r = 0. (2.3) D
36
Ji
Using theorem (2.1), we can prove some famous equalities which are known in [3] and also present some new equalities thai seem to be difficult if Ihey are built from geometric properties.
Proposition 2.1. With the above denoted, we have
(i) [3] ri + 7"2 + r3 = -I /? + r and riTa + r2r3 + r3ri ~ p'^.
... ri — rr2 — rvj — r r^ ~ r r^ — r r^ — r rii — r r^ — r r\ — r _ Tl + r r2 + r rg + r ri -\- r r2 + r r2 + r r^ + r ri+rr]-\-r AR AR AR. R r-irar,^
(iv) + + = 1 - 8 - + - ^ . Tl — r r2 — r r-i — r r r^
t \ 1 1 \ 1 1 1 1
(vi) 13] h^hb + hbhc + hcha = -^{rivi + r2r.^ + rgri). 2r ri
, ... {n-r2f , ( r . - r a ) ^ (rs - n)^ 4R (vu) 1 1 = o.
rir2 r2r-3 r^ri r
Proof, (i) From Vi, ra, r^ are three roots of the equation (2.3) and using Vieta's formula, we obtain
ri+r2 + r3 = AR + r and rira + r2r3 + r^^vi = p^.
(ii) From theorem 2.1 (iu), we have x^ — {4R + r)x^ +p^3: — p^r = [x — ri){x — i^2){x — rs). Then, choosing a; = r, we have
( n - r)(r2 - r){r3 - r) = iRr^ ^{^J.-i)(^±- i ) ( ^ - l ) = 4 ^ . ri — r (iii) Because r i , r 2 , r 3 are three roots of the equation (2.3). Let Vi = ,i =
Ti + r r(y, + 1)
1,2,3, we have Vi ¥= I and r^ = -^^ -. Substituting TI into the equation (2.3), we i--yt
obtain
{r"" +2Rr + p')yf -{- {2r^ + 2Rr - 2p^)y^ + {r^ ~ 2Rr +p'^)y,- 2Rr = 0,1 = 1,2,3.
T ^ s , yi, y2 and y^ are three roots of the equation
(r^ + 2Rr+p^)y' + ( 2 r V 2 H r - 2 / ) y ' + {r^ - 2Rr+p')y - 2Rr = 0. (2.4)
Using Vieta's i'ornnila, wc have
\Hr+r''--2Rr + p' ^ 2y,y2y:,+ !tr'l2 I !l2!h + !hlh - , . , ,^,,, ,_ ^,2 (IV) Because , , . 1 .. r,, arc three tools of the equation (2 3). we have
I 1 I _ : l i - 2 ( l / i ' 4 - r ) , ' 4_;<- _ T- r , ' ^ ,r ^ "*" .' - ^ " . ' • ' (4/i'4-^ ).'• + ; ' - ' - - / ' " ' • ' 1 ' 1 _ r ' - x / i ' i + p ' So Choosing .!• = r, we obtain 1 4- • ; — . 0 2
r i — r f.j — 1 13 ~ ' ^ ' ' ' 1/1' 1/1' _ l/i'_ ^ I ^R ^fjJVl r, ~ I 11- r I':, - r r r '
(v) Because r,, i^, / i are three roots of the equation . r ' - ( 4 W 4 - r ) . r - 4 - p ^ i - p V = 0, we deduce that —, —. — are three roots of p V j ' ' - ;r'.. - + I l/i' 4- r).r - 1 = 0 . Using
T\ r2 T3 Vieta's formula, we have
1 1 1 1 1 1 2 , , , J _ 1 1 1 1 1 - 1 .^4R + r i^ "'" ^ * r | " ' n ^ r^ ^ rs n '-2 ^ n r3 "*" r j r a ' H fr ' Otherwise, applying theorem 2.2 (ii) and Vieta's formula, we have
/ I 1 1 \ _ 2p' - 2r' - i<Rr _ 2 , 4 f l 4 - r H/;^ + / ^ ••" x f J " ^ '^r' ~ H " " p'r • / I 1 1 \ 1 1 1 1
Therefore, 4 7 : i 4 - 7 2 4 - r 3 = - 2 4 - - 4 - - T 4 - ^ .
\hl hi hfJ rf rr, i j r'
(vi) Applying theorem 2.l(ii, iii) and Vieta's formula, we obtain
• ' 5 - 1/ 2r
hahi + hihc 4- hch^ = r—^ = 2r— = -n(f\r2 + r2r3 4- r s r i ) . (vii) Applying theorem 2.1 (iii) and Vieta's formula, we have
( n - 1-2)' ^ (r2 - -4)-' , (r3 - r,Y ^ (4R + r)p' ^ ^ JR „ rir2 r2r3 r^ri p'r r
0 Next, we present how to use the transformation of rational functions to build new geometric equalities and inequalities. The form of transformation that is usually used here
ax 4- fe , , „, tsy = ;, a, 6, c, d 6 IR.
ex 4- a 38
Proposition 2.2. With the above denoted, let
^ ^ 3ri - 2r ^ 3-12 - 2r ^ 3^, - 2r 2ri 4- r 2r2 4- r 2/:i -!- r
^ 3r, - 2r 3i2 - 2r 3r2 - 2r 3r,i - 2r 3ri - 2r 3ri - 2r T^i = . . + - . , , „ , . 4--
^ 3
2ri H- r Ir, -{- r 2;'2 4- r 2r;i + r 2r;i + r 3ri - 2 r 3 r 2 - 2 r 3 r 3 - 2r
2T\ 4- r 2r2 4- r 2r;i f r We have
^ 3 r | - 2r .ir, - 2i' 3r3
2 r i 4 - r 2(2 + r 2r3 4 - r 19r2+12p'^' .il" — 2r 3 r(y + 2)
Proof. Let y, = —^ , 1 = 1,2, 3 we have 11, ^ - and ;-, = — ^ . From r i , r 2 , r 3
2a: 4- r ' 2 3 - 2y,
are three roots of the equation 2.3, so if we substitute r, into the equation 2.3, we obtain (3T' + 8Rr + 12p')y^ + (Ur' 4- 20Rr - 40p')yf + (8r^ - WRr + 39p')y,
-(4r^+4SRr + 9p^) = 0 therefore 1/1,1/2,1/3 are three roots of the polynomial in variable y
(3r^ 4- SRr + 1 2 p ^ ) / 4- ( l l r ^ 4- 20flr - 40p^)y' 4- (8r^ - WRr + 3Sp'')y -(4r^ + 4&Rr + 9p').
By Vieta's formula, we have
40p2 - 20flr - U r ' T l = t/i 4- s/2 4-1/3 =
T2 = !/l!/2 4- !/2!/3 4- J/3J/1
2 3 = !/l!/2!/3
3r' + SRr + 12p2 - 16i?r + 39p^
3r2 -1- 8Rr 4- 12p2 4r' + 4SRr + Qp^
3r^ + SRr + 12p^' 47.2 _|_ 4gj't -f- 9p2
Otherwise, let / ( t ) = — ^ — r r - ^ then / is an increasing function over [2r, +00].
Therefore, from if > 2r we have T3 > f(2r) i.e
3ri - 2r 3i-2 - 2r Srs - 2r lOOr^ 4- 9p^
2 r j 4 - r 2r2 4 - r 2r3 4 - r ^ 1 9 r 2 4-12p2'
D 39
Dam Van Nhi and Luu Ba Thang Proposition 2..V With the above denoted, lei
2r. - r ^21-2--I ^ 2
r, + I r. + r 2ri - i'2r^ (
I'l -\ I r2 \ r 2ri - r 2i'2 - r 2r
r3 I r lr2 ' • 2 / , , - r2 4- r r3 4- J
- 2 r i r3 4 r /i 4- r n 4- r 12 + r )•:, + r
then wc have an equality \l\ I /^2 - 2l\ = S Proof. I.ct ,(/,
'•('/, 4- 1) ^
1,2.3 we have if, ^ 2 and .r = — . From r,,r2,r3 mc lliK'c lociK nl llic ci|iuiliiin (2.3). so il wc substitute r, into the equation (2.3). we obtain
2(r^-l 2/i'r I / r ) ; / ; ' - 3 ( - r - 4 .'i;.')i/,- - 12(Kr -/i-');/, - (i" 4-8/?r ^ 1/r) = 0 Thcreloie, //i. //j. ,1/3 are three roots of the polynomial
2[r-\-2Ui +11')!/' -•.i[-r- + :iir)if - \2{lli --p')y - {r'+--III + lp').
Applying Vieta's formula for this polynomial, wc obtain 17 i 4- 72 - 21] — -j. D We know that elimination theory [ I ] is one of the most effective methods to solve polynomial equations, so if we combine this elimination theory and theorem 2.1, we can obtain some nice equalities. Hereafter, we give a brief summation about the resultantof two polynomials. Given a field K and two polynomials f.g€ L-CM] of positive degree
(ii).r'" + U] i'" ' 4- • • • 4- ((,„.ao ^ 0.m > 0 box" + feii""' -1 +b„,bo^O,n>0.
We build the square matrix of size (m 4- n) x (m 4- n) associated to f,g (called Sylvester matrix) as followings:
' ao ai • • • Urn
ao tt] • • • (j,„
S{f,g):={
fel •• fe,.
feo 6 1 • • • b„
(2.5)
feo fei • • • b„
where n first rows only depend on the coefficients of / , m last rows depend on the coefficients of g and die blank spaces are filled with 0. Then the resultant of / and g, denoted Res{/, g), is the determinant of Sylvester matrix S(f, g).
If we want to emphasize the dependence on i , we write Res(/, g, x) instead flt Res(/, j). Three properties of resultants [2, Chapter 3] are
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• (Integer Polynomial) Rcs(/..(/) is an integer polynomial in the coefficients o f / and 5 i . e R e s ( / , ( 7 ) € Z[ao,. ..f(,„./)(,. ...,6„].
• (Factor common) R e s ( / . g) = 0 if and only if / and g have a non trivial common factor in K[x].
• ( Elimination) There are polynomials IW e K[x] such that I' f + \'ii = H<'s(/'. </).
The coefficients of /". I" are integer polynomial in the coefficients of / and </
Proposition 2.4. (i) [ir +2Rr){lr + 2Rr){c^ -\-2R.r) = 2Rr{ab + bc-\-ca - 2Rrf.
(ri + / • . + /•, +/•)• a2+;y^ + ' l / ? r + r H ^ 4 - p ' ^ + 4/?r + / ' r ' + p 2 + -l/?r + r^
"•'•' " 5 = 5 5 5 Proof (i) Let f := x' - 2p.r' 4- (p' + 4R.r + r')x - 4Rrp = 0 and j := x' + 2RT - y.
By the elimination property, the two polynomials f(x),g(x) have a common root if and only if resultant R e s ( / , g; .r) = 0. So,
y^-(4p'' + 2Rr-2T)y'' + (T'-2R.T)y-2RrT' = 0 v/hixT = p'+ 2Rr + r'. (2.6) Leti/i = a^ + 2i?r,!/2 = fe^4-2iJr and 1/3 = c^4-2i?r. Because a, 6 and care three roots of f(x), thus yi, 1/2,1/3 are three roots of the equation (2.6). Using Vieta's formula, we have
yi!/2y3 = 2i?rT^ = 2RT(ab 4- fee 4- co - 2Rr)''.
(ii) Similar to the proof of (i), applying elimination theory [1] to two polynomials f(x) = x ' - 2pa:^ 4- (p' + 4RT + r')x - 4Rrp and 5(2:) =x'+p'' + 4Rr + r'-y, we obtain
j/3 _ {p2 + 4 R r -I- r ' ) l / ' - (2py - 4Rrp - 2r'pf = 0. (2.7) Let yi = a^ 4- p^ 4- 4Rr + r'',y2 = b'' ^-p' + 4Rr+ r' wiyi = 1? +p'' + 4Rr + r ^ then
!/i,y2.y3 are three roots of the equation (2.7). Using Vieta's formula, we have yiy2y3 = ( 4 R 4 - 2 r ) V V H e n c e ,
(0? + p2 + iRr 4- r2)(62 4- p" 4- 4 i i r 4- r'')(^ +p'' + 4Rr 4- r') = (4Rr + 2 r ' ) V - Otherwise, ri 4- r2 4- r3 = 4iJ 4- r, we have
(ri 4- r2 4- ra 4- r}^ _ a ' 4- p^ 4- 4Jir 4- r^ fe^ + / 4- 4Jir 4- r^ c' 4- p ' 4- 4Rr + r ' 5 S S 5 •
D If we combine them with the famous inequality R > 2r, we can find some interesting inequalities, for example
Corollary 2.1. Wilh Ihc above denoted, wc have
(a' I 2/,V)(fe-4-2/(r)(r'^-f 2 7 < ' r ) < 2 H "
Proof Let 7' = (rr 4- •lllr)(li' 4- 2 / ( / ) ( r - 4- 2 K r ) , by Proposition 2.4 we have (a^ + 2Ui)(lr I 2lli){r' \-2l!r) = 2lii {iili 1 lic + iii - 2/?r)-'. Ciiiiibiuing ii/)4-ic 4-ca < — 4p' and 1/r < 27/1*". wc obtain
/• < 2llr[<)ll' 2l!i ) - = ! • / • ^ «••' 2,(')/i' - 27 )-
Consider the function ,/(.i) i (')/i' .rf where 0 < .r ^ R. V^it have f'{.r) > 0 with each .r € (0, Ti*], deduce that the function / is the increasing function in (0. R\. Therefore /(•'•) < I(R) - II Iff'' with each X e (0, li]. Notice that R > 2r. so T £ MR'' and the
equality holds if and only if A / 1 / i f is equilateral. D Corollary 2.2. Suppose given a ciinvc\ quadrilateral .\li('P of .\B = u. BC =
b.Cn — c. P.) = d insciihed the circle of center (), radii R. Denoted r j . r 2 , r 3 , i] as the radius of the nu irclcs of triangles A.MU '. ABCD, ACD.4 and ADAB. respectively.
2''/f''
LetT = — ^ then ac + bd + 4.yr]r^r^
T > ^(ab 4- 4rf)(fec 4- 4n)(rd 4- 4r\){dii + 4 r ; ) . Proof Let .\C = .r. BD = y, by Proposition 2.4 we have
( a ' 4-27i!r,)(6^ -I- 2/?ri)(.i- -1- 2 « r i ) ^ MI^
Otherwise, we have (a' 4- 27?r, )(4^ + 2/?ri) > (afe + 2Rr,)'. so we deduce that ( o f e 4 - 2 « r , ) ^ ( . r ' 4 - 2 t f r , ) < 64R".
Similar, we obtain
(fec4-2;,'r,)-'(i/--F2yf/-,) < 647?"
( c t i - | - 2 / f r , ) - ( . r - F 2 7 ? r 3 ) < 647?' (da 4-2/?r,)•'(,'/•'4-2/?r,) < 647?"^
Multiplying sides to sides, we obtain
H''l>+^Iiri)(bc+2Rr2)lcd+2RrMda+2Rr,)Y{,-^ + 2Rr,){x''+2Rri)(y'+2Rr2){y^+2Rri)<2"R".
(2 8) We again have
(a:2 4-27?r,}(x='4-27?r3) ^ (x^ 4 - 2 7 ? 0 ^ f (!/'4-27?r2)(y V 27?r4) > ( y ' 4 - 2 7 ? v / ? 2 ? I ) '
and
{x' + 2RVnri){y^ + 2R^/iVi) > (xy + 2R^riT2T3Tif.
42
=> {xy + 2R^rir2r3r4)^ < (x^ + 2Rri){x'' + 2/^-,,)(/r ^ 2 / ^ r 2 ) ( / + 2/?r,) (2.9) Combining (2.8) and (2.9), we have
{ab-\^2Rrj){bc + 2Rr2){cd + 2Rr3){da + 2Rr4)i.rf/ |-2/?^r,'rar;,r^) < 2'^/?''-^ (2.10) From 2rt < R,i = 1,2,3, 1 and by Ptolemy's theorem, we have
xy + 2R-^rir2rsr[ = ac + bd + 2 / ? i y 7 v V V ^ > ac + bd + 4y'ri7^r^r^.
Therefore, we deduce that
^ , ^ 2'-'/?'^ ^ 2'^/?'^
(nr + fed + 4^rir2r3r,i)^ "^ (ac + bd + 2 / ? , y n r ^ r ^ ) ^
> (ab + 2i?ri)(6f + 2R.r2){cd + 2R.r^){da + 2/?r4).
Using again the inequality 2r, < /?,, ?- = 1, 2, 3,4, we obtain T^ yj{ab + 4r?)(6c + Arj){rd -\- Ar^)(da + h'i)
D As above, we have built some new interesting inequalities in a triangle in which we use the famous inequahty R > 2r. Similarly, we can obtain some more interesting inequalities as followings:
rf r, ri ri ro r^
- | + - | + ^ + - - - ^ 8 1 .
(^o(^o(^o^«•
(a-b)' 1 (fe-c)^ 1 ( c - a ) ^ ^ p^
afe ' fee ' ca ~ 4r2
1 1 1 1 1
27
•?'^(^0(p1-0(p1-0-^T-
The equaUties hold if and only if the triangle is equilateral.
The proofs of these inequalities refer the interested reader as exercises.
REFERENCES
[1] D. A. Cox, J. B. Little and D. O'Shea, 1998. Using Algebraic Geometry. Volume 185 of Graduate Texts in Mathematics. Springer-Verlag, NY, p. 499.
[2] D. Cox, J. Littie and D. O'Shea, 1997. Ideals, Varieties and Algorithms, 2nd edition.
Springer-Verlag, New York.
[3] D.S. Mitrinovic, J.E. Pecaric and V. Volenec, 1989. Recent Advances in Geometric Inequalities. Acad. Publ., Dordrecht, Boston, London.
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