Relatia Lui Stewart

ccccccccccRelatia lui Stewart c

c

ccccccccccccccc

cc
 c  
cc
 
c 
c 
c c c  cc  
c

c  c cc c  c
c
c cc

OA2× BC + OC2 AB - OB2 ×AC = AB× BC AC O

x A

Ʌ-x B

C


   
c c

AOB => OA2 = AB2 +BO2 ± 2AB× ×BO cosx | BC

½ ½

BOC => OC2 = BC2 +BO2 ± 2BO BC ×cos(Ʌ-x) | AB cos(Ʌ-x) = -cosx

OA2 BC = AB2 BC + BO2 BC ± 2AB ×OB BC

cosx

OC2

cosx

AB = BC2 AB + BO2 AB + 2BC

OB AB

OA2 BC + OC2 AB = AB2 BC + BC2 AB + BO2 BC + BO2 AB OA2 BC + OC2 AB = AB BC (AB + BC) + BO2 (BC + AB) OA2 BC + OC2 AB = AB BC AC + BO2 AC

Deci OA2 BC + OC2 ×AB ± OB2 AC = AB BC AC

(+)

OBS!  B

cc  AC,    AB = BC =

ÿ


 c
 cc
  OA2 BC + OC2 ×AB ± OB2 AC = AB BC AC => ÿ

OA2














=>OB2 =

+ OC2 - OB2 =

ÿ


- OB2 ×AC =



| : AC

ÿ 


 × 

ÿ

c
 c

c

A

mA

B

C

mA2 = mB2 =

cccccccccc


  ×

   × 


  × c
cc