20 13 the eighth problem of Guangdong one-mode mathematics

Prove: a+b+c = 180, 2b = a+c = 180-b, then b = 60.

Then we can know from the cosine theorem: cosb=(a? +c？ -B? )/(2ac)=cos60 = 1/2

That is (a? +c？ -B? )/(2ac)= 1/2

Answer? +c？ -B? = communication

Answer? +c？ =ac+b？

Answer? +c？ +ab+bc=ac+b？ +ab+bc

c(b+c)+a(a+b)= a(b+c)+b(b+c)=(a+b)(b+c)

[c(b+c)+a(a+b)]/[(a+b)(b+c)]= 1

[c/(a+b)]+[a/(b+c)]= 1

[c/(a+b)]+ 1+[a/(b+c)]+ 1 = 1+ 1+ 1

[c/(a+b)]+[(a+b)/(a+b)]+[a/(b+c)]+[(b+c)/(b+c)]= 3

[(a+b+c)/(a+b)]+[(a+b+c)/(b+c)]= 3

[ 1/(a+b)]+[ 1/(b+c)]= 3/(a+b+c)