设a,b,c分别是三角形ABC的
充要条件
1。先证a^2=b(b+c)是A=2B的充分条件
a^2=b(b+c)
4R^2sinA^2=4R^2sinB(sinB+sinC){正弦定理}
sinA^2=sinB(sinB+sinC)
(sinA-sinB)(sinA+sinB)=sinB*sinC
sinA-sinB=2sin[(A-B)/2]cos[(A+B)/2]
sinA+sinB=2sin[(A+B)/2]cos[(A-B)/2]
(sinA-sinB)(sinA+sinB)
=2sin[(A-B)/2]cos[(A+B)/2]2sin[(A+B)/2]cos[(A-B)/2]
=sin(A-B)sin(A+B)...全部
充要条件
1。先证a^2=b(b+c)是A=2B的充分条件
a^2=b(b+c)
4R^2sinA^2=4R^2sinB(sinB+sinC){正弦定理}
sinA^2=sinB(sinB+sinC)
(sinA-sinB)(sinA+sinB)=sinB*sinC
sinA-sinB=2sin[(A-B)/2]cos[(A+B)/2]
sinA+sinB=2sin[(A+B)/2]cos[(A-B)/2]
(sinA-sinB)(sinA+sinB)
=2sin[(A-B)/2]cos[(A+B)/2]2sin[(A+B)/2]cos[(A-B)/2]
=sin(A-B)sin(A+B)
sin(A-B)sin(A+B)=sinB*sinC=sinB*sin(A+B)
sin(A-B)=sinB
A-B=B
A=2B
得证
2。
证a^2=b(b+c)是A=2B的必要条件
很显然这题可以倒推,步骤大致为
A=2B
sin(A-B)=sinB
sin(A-B)sin(A+B)=sinB*sin(A+B)
。收起