Jump to content

Photo

Cho $\bigtriangleup ABC$. CMR: $abc(cosA+cosB+cosC)=a^2(p-a)+b^2(b-b)+c^2(p-c)$

* * * * * 1 votes

  • Please log in to reply
1 reply to this topic

#1
anhxuanfarastar

anhxuanfarastar

    Sĩ quan

  • Thành viên
  • 368 posts
Cho $\bigtriangleup ABC$. CMR: $abc(cosA+cosB+cosC)=a^2(p-a)+b^2(b-b)+c^2(p-c)$

INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE !!!


#2
Sagittarius912

Sagittarius912

    Trung úy

  • Thành viên
  • 776 posts

Cho $\bigtriangleup ABC$. CMR: $abc(cosA+cosB+cosC)=a^2(p-a)+b^2(b-b)+c^2(p-c)$

áp dụng công thức:
$cosA=\frac{b^{2}+c^{2}-a^{2}}{2bc};cosB=\frac{c^{2}+a^{2}-b^{2}}{2ca};cosC=\frac{a^{2}+b^{2}-c^{2}}{2ab}$
$\Rightarrow abc(cosA+cosB+cosC)=abc(cosA=\frac{b^{2}+c^{2}-a^{2}}{2bc}+cosB=\frac{c^{2}+a^{2}-b^{2}}{2ca}+cosC=\frac{a^{2}+b^{2}-c^{2}}{2ab})= \frac{1}{2}(a(b^{2}+c^{2}-a^{2})+b(c^{2}+a^{2}-b^{2}+c(a^{2}+b^{2}-c^{2}))=\frac{1}{2}(a^{2}(b+c-a)+b^{2}(c+a-a)+c^{2}(a+b-c))=a^{2}(p-a)+b^{2}(p-b)+c^{2}(p-c)$

(dpcm)

Edited by doandat97, 05-12-2012 - 20:30.





1 user(s) are reading this topic

0 members, 1 guests, 0 anonymous users