Plane algebraic curves drawn by the isotomic conjugate for a triangle : applications of a drawing tool and mathematica-香川大学学術情報リポジトリ

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Ｍｅｍ.Fac.Educ.、Kagawa univ. II、58(2008)、31-45

Plane Algebraic Curves Drawn

by thelsotomie Conjugate 11)ra7[¥iangle

−Applicationsof a Drawing lool and ]Vlathematica−

by

Keisuke M:ATsUMOTo，Kazunori

FUJITA alidHiroo FUKAlsHI

(Received May 30, 2008)

Abstract

ln this paper we present a cQmputer tool for drawing a locus of the isotomic conjugate to a point for a triangle which moves along a distinguished curve. The drawing tool provides many plane algebraic curves witb simple expressions. ＜

§1.1ntroduction

ln the sequel to [4]'[9]，[20]our

study aims

to develop

a drawing

tool on a

display for experimental

research on various curves using computers.

ln elementary

geometry

we have five significant notions for a triangle； that is，the

center of gravity，the center of an inScribed circle，the center of an escribed clrcle，the

circumcenter

and the orthocenter of i triangle，As a similar notion we have the isotomic

conjugate to a point for a triangle.

｀Whichcurve is drawl! as a locus of l;heisotomic

conjugate

to a point for a triangle

which

moves

under

a certain condition? ln this study we limit ourselves

to the case

where

a triangle is fixed while the point moves

along a distinguished curve. Then

our

main

concem

is to find various unknown

curves with simple algebraic expressions

as a

The last author was paltiallysuppolted by Grant-in-Aid florScientificResearch（No. 19500761）， the Ministry of Education， Culture，Sports，Science and Technology， Japan｡

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K.MATsUMOTO，K｡FU,IITA and H. FUKAlsHI

locus of the isotomic conjugate to a point for a triangle.For terminology of geometry

throughout the paper, consult[2]−[31，[11]，[13]，[14]，[18]and[22].

§2.A

program

for drawing a locus

On a plane let us consider a given triangle △Å召C andai)oint j)different from the vertices of tlle triangle.

Suppose lines Aj)，召j)，C7)intersect the opposite sides 6f a triangle △Å召C at points，￡)，￡，F and Z:)･s･E19F･ be the reflections of に)･，Ｅ，Ｆｉnthe midpoints ofthe corresponding sides. Then the lines λZ:)'，召￡'，CLF' always intersect at a point ？'， (or are

.pal‘allel，1.6ﾑ,intersectin the point of infinity (x))，calledthe j加zθz?zjccθ司昭αた to｣1)for 4AjC ； the map ψ :R＼{Å，j，(ﾌ)→RU{oo}defind by ψ(だ)＝j="is dalled the 泌応｢心明/昭d回for△Å斑]．

L6tA(s，r)，召(−1，0)，C(1，

0)be

the vertices of a triangle where

z≠O，and

？(M, 1')a

pOint different frOm

the vertlCeS. Then

the p01nｔＰ’(XI,

y)iS

determined

aS

thejntersection of two straightlines

jS￡'and

CF'

given by

・奉・・ｙ° ｙ° 召

バ雨＋,4)−(1+4)v}

z(1＋s)(1＋j)＋(3＋y)(1−j)y

バz(zj−1)＋(1−s)y}

z(1−s)(1−z,)＋(3−s)(1＋s)v

(い1)，

/

.

. 回忿￣胴曰ﾋﾞ=￣屁J 、（ﾚに1）

￡）ぴ Å Ｃ

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Plane

AlgebraicCurvesDIJawn by thelsotomic Colljugateflolj

a Tfliangle

Proposition. The

ise7ie

Qf the point Qf 垣紅吻，‘φ￣1(Qo)，jりμwzゆy涜ａ面pg

z2λ12−2m:y十G2十3)y2−2り'−z2＝O，

W11jdz芦z&s6法j'り叫冶哨Ewrrjc6Å,召，Cαz

｢＆z･μ/zEみαzでycazMrげ△Å召Cα･sj￡scEnたz:

j)roo:f1 Let

7:)be apoint

such

that ψり))゜oo.

The

slope

s･･ of the linφjS￡'is

equal to the s16pe j恥

ﾚ

of the line F'C. 万

＼犬

Find

the Groebner

bases (see[11，[12])by

applying

Mathematica

ver 6.0by

M/olfram Resea:rch lnc･，to the following code :

1 U:゜X

v:゜y

fl:＝(t＊(1＋s)＊(1+u)+v(3+s)＊(1−s))mb-t＊(tftu-v-sv)

f2:＝(t＊(1−s)＊(1-u)+v(3-s)＊(1＋s))mc-t＊(-t†tu+v一sv)

f3:=mb-mc

Gro9bne7｀Basis[{f1，f2,f3}･,{!nc,mb,t，s,y,x}]

Factor[Z]

Then we obtain the term

z(−r2十＆j2−2印−2my十3y2十jy2)

in thelistof the generating Groebner bases. Therefore, we have

(＊＊) ゐ

;

2−

2m

E

y十(3十s2)y2−2印−z‰0

as anequation of ψ￣1(oo)whichpresents the ellipse,because

r2(3＋s2)−Gz)2＞o.

Setting the leftside of (＊＊)byﾊﾟyy)，we

have the following:

・

１

)The

emPse

(＊＊)passes through the vertices of △Å召Ｃ，

ii)The

ellipse (＊＊)has the point

of the equations

が

一−

∂,χj

汗

ﾙ)as

its center, which is given by the solution

O and

が

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K.MATsUMOTo,K｡FU,IITA and H. FUKAlsHI

Which

curve is drawn

as a locus of the isotomic

conjugate

7）’

to a point， 7）for

△AjSC

which

moves

along a distinguished curve

y

？

We

divide our operation of a drawing tool into the drawing part and the printing

part,due to the circumstances of our computer machines.

PART

ONE : To

draw

a locus of the isotorilicconjugate to a point for a given triangle.

A program

of our drawing

game

for Case

l of Theorem

ia § 3 is written in visua1

B asic Ver. 6.0 by Microsoft

Corporation

and consists of the following

five st6ps ｀(see

List l).

Step 1.

Step 2.

Step 3.

Step 4･

Step5.

Set the coordinates axes and triangle △Å召C in black.

Set the initialpositk)n of a point ？on

curve

ｰ(in blue)，

Draw

the ellipse that pre帥ntsψ'1(邨)in

yellow ocher.

Plot the iiotomic conjugate j)'in red.

ニ

When

one moves

the point j)continuously

along the curve

ｰ by the mouse。

the point j)'continuously

draws a locus ，y with a solid line in red.

犬

PART Two

: 7ITo process a bitmap file(bmp)bypびIEX2ε'to

exhibitit on another

display,an4 to printit.

OUTLINE

of the PROGRAM. Let

A (0，2)，召(−1，0)，C(1，0)and

y:y＝た.Where

a

pOint戸(Z4,1ﾉ)On

ｰiS

SeleCted by the mOUSe,

then the iSOtOmiC

COnjUgate

7)'(XI,

y)to

■ I ・ ●

？is deter面ned

by the equations (＊)for

G，r)＝(0,2).

The program

for drawing

a locus y

of the point 7)'is given in List l. ln thiscase

we win have the curve

y: 4む2−8xy十3げ−4肪一計＝０

on a display (Fig. 1).

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Å(0,2)，j(−1,0)，C(1,0)，

yレｙ＝た(--？十1)

犬

(λ

;

≠O)レ彪卯

｢･必ｚ／

ｙ：４(た−2)x2−3(3た＋2)y2＝4(3た−2)y−4た＋８＝ｏ：

i)the

straightline

ii)，the ellipse

iii)the parabola

iv)the

hyperbola

忙ド

{Ｊ，た≠−1,(い

forん＝2，

orん＞2.

1･4（づ

for −でF＜ん＜2， forん＝−{，

Plane Algebraic Curves Drlawn by thむlsotomic Co哨ugate f6r a Triangle

§3. Algebraiccurvesobtained

as a locus of the isotomic conjugate

As a locus of the isotomic

coiljugate to a point for a given triangle we have varioui

curves. ln

this section

we exhibit some

of them

together

with

the cases of plane

algebraic curves

with simple expressions

WhoSe

graphs

are not given in the standard

texts of the subject [101，[121，[151，[171，[191，[21]and[251.

Theorem，￡α油Jげがz･りＷ／θｗ訥gαな訪Γαjccμn.?Es 。ya,H,ε涌zαj,z 气ｿHz勿csげ

the isotomic conjugate l）’Zθαjpθ訥Z？知r ･2 rrjα凧grZE△Å召C w/zjc/1謂θW･∫α/θzlgが1･E! czzr1.ぞｰ。：下

Case l (Fig.1)

Å(0,2)，S(−1,0)，C(1,0).

ｰ: χI＝k，the stmight nne，

y:

4尨ヲー8,xy＋3戻−4砂−4灸＝o

thatis， '

i)the

straightline f()r匹−1, 0，1，

ii）the ellipse

iii）the parabola

)V)｀the

hyp&bohi

Case 2 (Fig.2)

that is， for

_た＜/IJ

_or

forλ

;

゜±{J，

f｡fづﾆ{みｄｘ

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K.MAT｀sUMOTo，K.FU,JITA and H. FUKAlsHI

Case

3 (Fig.3).

Å(0，−3)，j(−1,0)，C(1,0). .j

犬上

‘ぎ: y°.χ?−4，哨εραn2みθZα.

y

:

5y4十74y3十２(ｘ２十5)y2十48(x2−1)y十9(x2−1)2＝0

Case 4 (Fig.4). .･

Å(0，−1)，j(−1,0)，C(1,0).

y:＼χ12−y2°−1，thehyperbola，

y : y3−y＋ (3×2−1)y−(x＋1)(x−1)＝o

Case 5 (Fig.5).

Å(0,3)，j(−1,0)，C(1,0).

ｰ: ，x2十(y−1)2＝1，向d

｢ε.

y:

3y4−28y−2(7×2 − 45)y2十108(x2−1)y＋27(x2−1)2＝0

Case

6 (Fig.6).

Å(0，1)，j(−1，0)，C(1，0).

十

y

:，x2十(y十1)2＝4，向ｄ

｢ε．

ｙ：２:y3 − 3y2− (x＋1)(x−1)＝o. ◇

j)n9げ Let れz4,V)aj j)″(XI,y)･

エ

Case l. Find

the Groebner bases by applying Matheinatica Ver 6.0 to the following code :

S:=0

u:=k

fl:＝(t＊(1＋s)＊(1+u)+v(3+s)＊(1−s))mb-t＊(t+tu-v-sv)

f2:＝(t＊(1−s)＊(1-u)+v(3−s)＊(1＋s))mc-t＊(-t+tu+v-sv)

f3:゜y-mb*(x゛1)

f4:゜y-mc*(x-1)

GroebnerBasis[{fl，f2,f3,f4}･，{mc,μ},v,k,y,x}･]

Factor[Z]

Then we obtain the term

r(一尨2十kty)cl− 2抑−４吋十３が)

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Plane Algebraic Curves Drawn by the lsotomic Co哨ugate for a Trliangle

奴x2− 8巧十３砂2−4妙−４ん＝０

as an equation of ，y.

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K.MArsUMOTo、K.FU､IIIAand H. FUKAlsHI

References

[111V.MK.Adams and R Loustaunau : Azz 7z2zn7,迦czj∂zzz∂G屁丿&詔r＆z認s，Graduation Studies in Math.，VOI.3，Amer.Math.Soc.，Providence，1994.

[2]H.S.M.Coxeter : 佃zyり＆4czjθn応Gg∂撰Erり･，2nd ed.，John W711ey and Sons lnc.，New York，1980.〔コクセター(銀林浩・訳)：幾何学入門第２版，明治図書，東京，

1982.〕

[3]H.S.M.Coxeter and S. L.Greitzer : GE∂z7zEzりり&yj認j，New Mathematica1 Library， Number 6， SchooI Mathematics Study Group， Random H(juse， lnc･， New York， 1 967 ｡〔コークスター，グレイツァー(寺阪英孝・訳)：幾何学再入門，SMSG双書，

河出書房新社，東京，1り70.〕

[4]K. Fujita， K. Matsumoto and H. Fukaishi : Plane Algebraic Curves Drawn by the lsogolial Conjugate for a TrianglQ − Applications of a Drawing Tool and Mathematica， Mど謂｡瓦zc.￡d14c･，￡2gαwαび㎡wrsj印/7，58(2008)，1-14.

[5]K.Fujita and H. Fukaishi :･ Plane Algebraic CurvesDrawn by the Orthocenter of a Peda1 Trianlille−Applicationsof a Drawing lbol and Mathematica， j＆1?zθj肖げz/zEjl;包c ｢印げ￡＆a?αzj∂zz,瓦agαwαびzijwrj印弑57(2007)，51-72.

[6]K.Fりjita, A, Matsushima and H. Fukaishi : A Locus of the Orthocenter of a Triangle − lnstruction in Geometry by a Moving Locus on a Computer， Ma?1. j42c. ￡＆4c･，瓦αgawα IL/zijwrsjり,j7，55(2005)，1-13.

[7]K， Fu,jita，A. Matsushima and H， Fukaishi : A Triangle with Three Distinguished Collinear Points − lnstru(jtion of Geometry by use of a Drawing Game on a Display， Ma?z.瓦zc,￡＆!c･，瓦･2g･2wα￡ﾉ7㎡w肖jり･･j7，55(2005)，25-41.

[8]K.Fujita,A. Matsushima and H. Fukaishi : A Triangle with Distinguished Concy(;lic Points − lnstruction of Geometry by use of a Drawing Game on a Display，

j/1＆z?z.Rzc. ￡jz4c｡￡2gαw･2ひzjwz″siり･77，56(2006)，1-25.

[9]H. Fukaishi， K， F1!jita and A. Matsushima : Altemative Geometric Proofs of Theorems for Cohcyclic P()ints for a Triangle， Ma?z.瓦zc.￡＆zc･，瓦αgαwαび㎡wz°∫jり･j7，56(2006)， 51-59.

[10]IV. Fulton : ﾒ11なEZ･Mjc CMn'6jV. A. Benjamin， New York， 1969.

[11]羽鳥裕久：数学の小さな旅，近代科学社，東京，1992.〔＝H.Hatori : SmaU Travds 訥Ｍ･2がza?zazia， Kindai-kagaku-sha， 'lokyo， 1992.〕

[12]一松信・他：新数学事典，大阪書籍，大阪，1979.〔＝S.Hitotumatu et al.:Shin- Saga㎞-j7zgrz， 6saka-shoseki， 1979.〕

[13]一松信：代数学入門第三課，近代科学社，東京，1994.〔＝S.Hit(jtumatu :

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Plane Algebraic Curves Drawn by the lsotomic Cor!jugate for a Triangle

［14］一松信：現代に活かす初等幾何入門，岩波書店，東京，2003.〔＝S.Hitotuμ1atu :

Gendai ni l㎞su Shotakika N･yi??M)i1，lwanami-shoten， lokyo， 200幻

［15］飯高茂：平面曲線の幾何，共寫出版，東京，2001.（＝S.litaka : G･･・吻yげ

jl）1どzzzgC￡4n’6,Kyorits1!-Shuppan, lokyo, 200 1 ｡〕

［16］石谷茂：数学ひとり旅，現代数学社，京都，1998.〔＝S.lshitani : S卯α辿

Hitoritabi，Gendai-sngaku-sha， Kyoto， 1998.〕ニ

［17］河田敬義：代数曲線論入門，至文堂，東京，1971〔＝Y.Kawada : jj･zr, 歛･aza

Aな訪r ｢cC岬″w77zEθり4 Shibundo/Tokyo， 1975.〕

［18］小林幹雄：初等幾何学，共立出版，1958.〔＝M.Kobayashi : ￡laiaaり･GEszg咋 Kyoritsu-Shuppanjn〕kyo， 1958.〕 ∧

［19］E.Kunz : 7nrrθ,仙czj∂nz∂j）1αz2εÅなεゐΓαjcCMryEs，Birkhiiuser，Boston，2005. （Translated from the original German by R. G. Belshoff.）

［20］A.X/latsushima，K.Fりjita and H. Fukaishi : A Locus of the Orthocenter of a Pedal Triangle − lnstruction of Geometry by use of a Drawing Galne on aDisplay， jMEz?i.R2c. Educ･，瓦昭αｗα防2jw尽岬弑57（2007），1-15.

［21］G.0rzech and M. 0rzech : j）1,2,1･Ålg訪,㎡cCz,n,a，Marcel Dekker， New｀Ybrk，1981. ［22］D.Pedoe: GEθz旭rzでy，Dover Publ. lnc.，New York， 1970.

［23］坂井忠次：グラフと追跡，培風館，東京，1963.〔＝c.sakai: a,φほ,乃函函， Baifukan，Toky0，1963.〕

［24］Yvonne et Ren6 Sortais ZLα G＆・＆･jど面7Maがε，Hermann，1987.〔＝YIソルテー，Ｒ．ソルテー（戸田アレクシ哲・訳）：なぜ初等幾何は美しいか，東京出版，2002.〕

［25］R. J. Walker : Åな訪r ｢c CMn’Es， Springer-Verlag， New York， 1950.

Keisuke MATsUMOTO

9-121-2 lkeda-cho/lajimi-shi，Gifu，507-0048，JAPAN

Kazunori FUJITA

Depaltment of Mathematics， Faculty of Education， Kagawa university 1-1 Saiwai-ch0， 111kamatsu-shi，Kagawa，760-8522，JAPAN

￡-z7zd 殍,as:fujita＠ed｡kagawa-u｡ac.jp

Hiroo FUKAlsHI

Department of Mathematics， Faculty of Education･ Kagawa university 1-1 Saiwai-cho，Takamatsu-shi，Kagawa，760-8522，JAPAN

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Ｃａｓｅ

Case

ん＝１

ん＝

且

K.MArsUMOTo,K.FUJITA and H. FUKAlsHI

ＣａｓｅＣａｓｅ左＝2.300 た＝0.667 Fig. 1 （ｰ y

λ7＝ん

4旨七8ぞ十3似七4似一収＝０

(11)

Case ん＝２

Ｃａｓｅん＝

Plane Algebraic CurvesDrawn by the lsotomic Conjugate for a Triangle

２ −_３ Fig. 2 貿ｙ Case た＝−0.300 Case 灸＝−3.000 ｙ４＝ん（ _{ｙ十げ（ん≠o）} (ん−2)y−3(3ん＋2)ダー4(3ん−2)y−4ん＋８＝ｏ

(12)

Fig. 3 貿ｙ

Fig. 5 貿ｙ

K.MxrsUMOTo,K.FUJITA and H. FUKAlsHI

jにぞ−4 5戸＋24y3十２(ぞ十5)戸十48(x2−1)y＋9(x2−1)2＝0 Fig. 4 貿ｙぞ亘y−1)2＝1 3ダー28ダー２(ﾌｽﾞ2− 45)ダ十108ヅー1)y十27(y−1)2＝0

ぞーゾニー1

ダーダ十(3y−1)ﾊﾞ(x十1)(x−1)＝o

Fig. 6 貿ｙぞ十(y十1)2＝4 y−3ダーG＋1ぬ−1)＝o

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- 一４ ∼

Plane Algebraic Curves Drawn by the lsotomic Co哨ugate f6r a Triangle

List 1. A prggram

for drawing a locus

一 -ll● ｜ -pse φ'1((x)) 一一 -∼

that presents

Specifying statements

of the gineral viiriables

Dim

SX, sy,ex,ey

As Double

Dim

ax, ay, bx，by, cx，cy, dx, dy As Double

Dim

qx, qy As Double

Dim

px As Double

Drawing

the initial figure and the el

Private Sub Form_Activate()

Forml.AutoRedraw＝True

FormLLine(sx，0)-(ex，0) Forml.Line(0，sy)-(0，ey) h＝0.06

For x ＝lnt(sx)To lnt(ex) Forml.Line(X，−h)-(X，h) Next x

For Y ＝lnt(sy)To lnt(ey) Forml.Line(-h，Y)-(h，Y) Next Y

Forml.DrawWidth＝2 ｀Forml.Line(ax，ay)-(bx，by) Forml.Line(bx，by)-(cx，cy) Forml .Line (cx，cy)-(ax，ay)

pi＝4 ・ Atn(1)

For t ＝O To 2 i pi Step 0.001 ｡･

レ d･＝3・(ayべ2 ・ Cos(t)A 2 -･ 2 ・ ax t ay･1 Cos(t)・Sin(t)十(axA 2･十 3)1･ Sin(t)･A 2) r＝Sqr((4・ayA2.)/d) ×＝ax/3十r yCos(t) 。＝ Y＝ay/3十r ・ Sin･(t) Forml.PSet(×,Y)，QBColor(6) Nextt Forml.AutoRedraw＝False Forml.DrawWidth＝2 Forml.Line(2，sy)-(2，ey)，vbBlue Forml.DrawWidth＝3 px＝2 Textl.Text＝px Locus End Sub

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１６Ｓ６ａｘｂｘ１ - 一一４ｇＳＳ

-K.MAT｀sUMOTo，K.FU,IITA and H. FUKAlsHI

- 一一 - - 一一

一一一･一一

-Setting the form， the coordinate axes and the initial

values of the coordinates of each vertex of a trian9le

O: −1

Private Sub Form_Loado

Forml.T6p＝500

SX＝-4.8

ex＝4.8

WX＝ex

− SX

wy＝wx

t Forryil.ScaleHeight / Forml .ScaleWidth

sy＝-0.51wy

ey＝0.5吟wy

づ

Forml .Scale(sx，ey)-(ex，sy) .･

Forml.BackColor＝VbWhite

CX＝1: Cy ＝OI

Textl.FontSize＝16

End Sub

Calculating the intersection of two lines

qy＝（rD･ t2 − r2 り:1）/d End Sub

-Private Su b.lntersection(xa,ya，xb，yb，xc,yc，xd，yd)

rl＝ya

− ylb:sl ＝xb − xa: tl ＝ya゛xb−xa

゛yb

'

r2＝yc

−yd: s2 ＝xd−xc:t2

°ylc゛xd−xc゛yd

d＝rl

●s2 − r2･きsl

qx＝(tl ・ S2 − t2・ SI)I/ d･

left ｂｕｔｔｏｎｏｆｍｏｕｓｅ

Action corresponding

to the

Forml.Line(X，sy)-(X，ey),vbBlue Forml.DrawWidth＝3 px＝X Textl.Text＝lnt(1000 1 px)/1000 Locus End Sub

Private Sub Form_MouseDown(Button

As lnteger, Shift As lnteger, X As Single,,YAs Sin9le)

Forml .Cls .

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ｇ

Ｓ

Plane Algebraic Curves Drawn by the lsotomic Coりjugate for a Triangle

- ･

Presentin9

一一

-of

_{the isotomic conjugate}

Private Sub Textl_DblClicko

px＝val(TextLText)

Forml.Cls

Forml.DrawWidth＝2

Forml.DrawWidth＝3

LOCus

End Sub

Private Sub Disp(py)

lntersectionax，ay，px， py， bx，by，cx，cy dxl ＝qx: dyl ＝･qy dx2＝cx − dxl 十bx dy2＝cy−dyl +･by ･･ lntersection bx， by，px，py，cx，cy，ax，ay exl ＝qx:.eyl＝qy ex2＝ax−e)tl十cx ey2＝ay − eyl 十cy

lntersection ax， ay，dx2，dy2， bx， by， ex2，ey2 pdx＝qx: pdy ° qy

Forml.PSet(pdx，pdy)，vbRed End Sub

Private Sub Locu50

For Y ＝50 t sy To 3 ゛sy Step O.1 Disp Y.

Next Y

For Y ＝3 ・ sy To 3 ･i ･ey Step 0.001

Disp Y ▽

Next Y

For Y ＝ 3 ・ey To 50 1･ ey Step O.1

1 Disp Y ･，

Next Y

End Sub

Plane algebraic curves drawn by the isotomic conjugate for a triangle : applications of a drawing tool and mathematica-香川大学学術情報リポジトリ