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

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• 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)

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

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