海龍公式專輯math.ntnu.edu.tw/~horng/letter/904.pdf · 網址: ??horng 海龍公式專輯...

Download 海龍公式專輯math.ntnu.edu.tw/~horng/letter/904.pdf · 網址: ??horng 海龍公式專輯 海龍公式專輯

Post on 13-Sep-2018

217 views

Category:

Documents

1 download

Embed Size (px)

TRANSCRIPT

  • HPM

    Heron Metrica

    --

    p

    ()

    1998 10 5 5 http://math.ntnu.edu.tw/horng

    HPM

    HPM

    (2005 ) 68

    2( )( )( )p a p b p cr

    p

    =

    1 ( )2

    a b c= + + r 2p

  • HPM

    vs.

    (1)Heron Metric (2)

    (1)--(2) (3)

    HPM (1) (2) (3) (4)

    ])2

    ([41 222222 bacac +=

    a b c> >

    HPM

  • HPM

    Heron Metrica

    Heron HeronHero of Alexandria 175(

    ) 350 (Alexandria)

    MetricaDioptra

    http://www.mlahanas.de/Greeks/HeronAlexandria.htm Heron of Alexandria

    MetricaMetrica

    Metrica 133

    3~12

    3 100

    Metrica

    720

    Metrica

    47[Euclid I-47]

    =21

    8

    http://www.mlahanas.de/Greeks/HeronAlexandria.htm

  • HPM

    789

    122/2424987

    ==++

    391248125712

    ===

    720324024046060512

    ===

    720=

    If A is the area of a triangle with sides a, b and c and s = (a + b + c)/2, then A2 = s (s - a)(s - b)(s - c).

    rsrcbarcrbra =++=++ )2

    (222

    s r

    AB AB ABC ABC

    ABBCCA (c, b, a ) ABC[Euclid IV.9] DEF G

    AGBGCGDGEGFG [Euclid I.41] BCEG2BGC

    CAFG2AGC ABDG2AGB

    2ABC EG() [Euclid III.17]

    BC BHAD CBH ABC AD=AF, DB=BE, FC=CE

    CHEG 2 2CH EG ABC ABC= = i 2

    G B

    FD

    E H

    A

    C

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

    )()()()(

    rsGECHABCBEcsBHbsCEas

    CHs

    =====

    =

    B

    F D

    A

    C

    K

    G

    E H

    L

  • HPM

    ))()(()()( 22 csbsassrsABC ==

    B BC (CBL90)

    G GC (CGL90) L LC BC K

    [Euclid III.31] CBL CGL CGBL , CL [Euclid III.22] CGB+CBL180

    CGB+AGD180 AGD= AGFBGD= BGFCGE= CGF

    AGDCLBADGCBL(90) AGDCLB(AA ) BCBLADDGBHGEADBH, DGEG [Euclid V.16] BL GE ()

    BCBHBLEGBKKE [Euclid V.18] BC+BHBHBK+KEKE

    CH BH BE KE CH2CHHBBEECEKCEBEECEG2

    CH2EG2(CHHB)( BEEC) 2ABC

    CH, HB, BE, EC ABC Q.E.D. [Euclid]

  • HPM

    12021261

    1244 1247

    13 14 15

  • HPM

    300 315 2

    4

    1 240

    a b c

    ])2

    ([41 222222 bacac +=

    84])2

    141513(1513[41 222222 =+= 315 ?

    240()

    315

    10024030030084

    =

    a h x (a-x) x c

    2222

    2

    2)

    a-bac-(ch +=

    21

    ])bac(a[c)a

    -bac-(caah 2222

    222222

    2

    241

    221

    21 +

    =+

    ==

    c bh

    x a-x

    2222 -(a-x)b-xc =a

    -bacx2

    222 +=

    222 -xch = 2222

    2

    2)

    a-bac-(ch +=

  • HPM

    (, 1997

    ()

    2 2 2 2 2 22 1 a +c -b a +c -b= ac+ ac-

    4 2 2

    2 2 2 2 2 22 1 2ac+a +c -b 2ac-a -c +b=

    4 2 2

    2 2 2 22 1 (a+c) -b -(c-a) +b=

    4 2 2

    ( )( )2 b+(c-a) b-(c-a)1 (a+b+c)(a+c-b)=4 2 2

    ( )( )2 b+c-a a+b-c(a+b+c)(a+c-b)=2 2 2 2

    ( ) ( )b+c-a a+b-c(a+b+c) (a+c-b)=2 2 2 2

    2

    a b cs + += a+b-c =s-c2

    a-b+c =s-b

    2

    b+c-a =s-a2

    = ( )( )( )s s a s b s c

  • HPM

    --

    (Heron 10 75 )

    16 17

    10

    4 5

    7912 13

    1821

    c b

    a CB

    A

    a,b,cABC 1 ( )2

    s a b c= + +

    ABC ( )( )(s s a s b s c )

  • HPM

    1. A

    Cxx

    oo

    I

    D

    EF

    B( )

    I ABC

    ID IE IF r= = =

    BDI BFI CDI CEI AEI AFI

    BD BF = CD CE= AE AF=

    AB AC BH CE= CK BF= A

    B Cxx

    oo

    I

    D

    EF

    HK

    G( )

    AK AH s= =

    AI GGK AK HG

    AHG AKG HG KG =

    ( AK AH s= = AG AG= ) HAG KAG =

    2.

    BM CD= GM

    HP BD= PG

    PG CG=

    0

    90

    PHG CKG

    HP CK HG KGPHG CKG

    PHG CKG PG CG

    = = = = =

    A

    B Cxx

    oo

    I

    D

    EF

    HK

    G

    M

    P( )

    CMG PHG

    (CG PG= CM PH= MG HG = )

    090CMG =

  • HPM

    BHG BMG BGH BGM =

    090BHG BMG = = 0180MBH MGH + = 0180MBH DBF + = MGH DBF =

    1 12 2

    BGH MGH DBF FBI = = = A

    B Cxx

    oo

    I

    D

    EF

    HK

    G

    M

    P

    BGH IBF

    : :IF FB BH HG=

    IF HG FB BH =

    AFI AHG

    : :IF HG AF AH=

    2

    : :IF IF HG AF AH =

    2: :IF FB BH AF AH =

    2

    AF FB BH AH IF =

    2 2

    AH IF AH AF FB BH =

    1 ( )2

    AH s a b c= = + + IF

    AF s a= FB s b= BH s c=

    2 2 ( )( )(s r s s a s b s c= )

    ABC sr ( )( )(s s a s b s c )

    CMG PHG CG PG=

    CM PH= MG HG= CMG PHG

  • HPM

    1634

    16 1643

    090CMG =

    (1633-1721) 27

    14 1675

    12

    22

    12

    090CMG =

    1.

  • HPM

    2.

    CM CK= BM BH=

    AK N KN BH= A

    B Cxx

    oo

    I

    D

    EF

    HK

    G

    M

    PN

    AH P HP CK=

    CN BP BC= =

    CG BG NG PG

    CKG PHG CK HP=

    090CKG PHG = = KG HG=

    CG PG =

    NKG BHG NG BG =

    A

    B Cxx

    oo

    I

    D

    EF

    HK

    G

    M

    P N

    NCG BPG BCG BPG BCG =

    MG

    HP CK CM= = CG PG=

    PHG CMG CKG PHG CMG CKG 090CMG =

    3. I

    090CKG CMG IEC IDC = = = = 0180MCK MGK + = MCK DCE + 0180=

    MGK DCE = MCK DIE = I

    IEC CKG ()

    : :IE CE CK GK =

    CE CK IE GK = 3

  • HPM

    AKG AEI

    : :IE GK AE AK=

    2

    : ( ) :IE IE GK AE AK = 4

    34

    2: ( ) :IE CE CK AE AK =

    2 : ( )( ) ( ) :r s b s c s a = s)2 ( )( )(sr s a s b s c =

    CMG 3

    BGH IBF I

    5

    40

    17

    (

    )

    xx

    o o

    CMG()

  • HPM

    : :=

    =

    2

    2 2

    : : ( ) : =( ) := =

    2

    : ( ) :=

    2: ( ) ( )( ) :s s a s b s c r = a= b= c=

    2 ( )( )(sr s a s b s c= )

    CMG

  • HPM

    18111882

    A. Wylie18551887

    (De Morgan, 18061871) (E. Loomis, 18111899) (Joseph Edkins)

    (Herons Formula)

    QA

    2.1

    ABC abc

    s 1 (2

    s a b c)= + +

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

    ABC I ID IE IF I

    r 1 ( )2

    a b c r sr+ + =

    b

    a

    c

    B

    A

    C

    RHS AIE AIF, BID BIF, CIE CID

    AE AF,BD BF,CD CE= = = zCECDy,BFBDx,AFAE ======

  • HPM

    FI

    E

    A

    r

    zy

    y zB D

    C

    xx

    1 (2

    s a b c)= + +

    xyz s-ax = yz yz (s-b)(s-c)= 2r

    s

    2( )( )( )s a s b s c sr =

    22))()(( rscsbsass =

    ( )( )(s s a s b s c ) 2.2

    2( )( )( )s a s b s c sr = 2xyz sr=

    2yz sr x

    = AH hAH

    ABH ACH A

    DH

    r

    B

    I

    C B

    NM h

    EF

    C

    I ABAC BC 'B '

    C

    ABH~ IB'D ACH~ IC'D AA

    (AB BH):(IB' B'D) h:r+ + = ( AC CH):(IC' C'D) h:r+ + =

    (AB BH):(IB' B'D) (AC CH):(IC' C'D) + + = + +

    (AB BH):(AC CH) (IB' B'D):(IC' C'D) (*) + + = + +

  • HPM

    I BC AB M AC N

    MI //BB' MBB'FMI = MB//IB' ' IB'DMBB =

    IB'DFMI = 0B'D , DB'=90 ,FMI I IFM I IF ID r = = = =

    ' (AA ) 'IFM IDB S IM IB =

    'BMIB 'IM IB= 'BMIB

    ' 'IB BB= IC' CC'= IB'D

    IB' B'D BB' B'D BD+ = + = IC'D IC' C'D CC' C'D CD+ = + =

    (*) (AB BH):(AC CH) (IB' B'D):(IC' C'D) BD:CD :zy+ + = + + = =

    2.3

    (AB BH) : (AC CH)= :y z+ + AB BH AC CHy z+ +

    =

    c BH b CHy z

    + +=

    c BH b CH hy z

    + += =

    r

    2( )c BH b CH hy z

    + + =

    r

    2

    2 (**)( )( )yz r

    hc BH b CH =

    + +

    2: ( )( ) :yz c BH b CH r h + + = 2

    ABH 2h22 2 ( )( )h c BH c BH c BH= = + (1)

    ACH 22 2 ( )( ) (h b CH b CH b CH= = + 2)

    (1) (2)

    (1) 2

    (3)( )( ) ( )( )

    yz rc BH b CH c BH c BH

    = + + +

    (2)

  • HPM

    2

    (4)( )( ) ( )( )

    yz rc BH b CH b CH b CH

    = + + +

    (3) (4)

    (3) ( )(c BH b CH+ + ) ( )(c BH c BH + )

    ( )c BH+ (c BH ) ( )b CH+ ( )c BH

    (b CH+ ) (c BH )

    (4)

    (1) (2) ( ) (( ) (b CH c BHc BH b CH+ +

    =

    ))

    2.4

    ACH ABH

    ( ) (( ) (b CH c BHc BH b CH+ +

    =

    ))

    =

    ( ) ( ) 2b CH c BH BC b c a b c s+ + + = + + = + + =

    ( ) ( ) ( ) 2 2( )b CH c BH b c BC b c a a b c a s a x + = + = + = + + = = 2 2OP s a b c= = + + OP S 2PS x= R OP

    R OP 'RR 'R 'RR OR= 'OR PQ

    Q ' ~ORR OPQ 'ORR OPQ

    VSR

    U

    T V

    T

    R

    P S

    U

    O Q

    'TT OP 'RR OP 'ST T 'ORR

    OPR

    'ST T ' 'ST T T= 'PT T 'PRR R

  • HPM

    OP 'OR PR RR s= = = 'PRR

    'PT T

    ' 'PT T T= ' 'PT T T ST= = '

    1 (2

    s a b c)= + +

    PR OR 'RR =

    2PS x= ' 'PT ST= 'PT 'ST

    x sx

    = =

    OP U U OP 'U 'PU 'SS V

    'OUU OPR 'OUU

    'OU UU=

    PU OU 'UU V OP 'VV OP

    'V

    'SV V 'OUU OPR

    'SV V

    ' 'SV V V=

    ' '' ~ '' '

    PU PU UUPU U PVVPV P