ΔABC a^2=b(b+c)证A=2B

来源:学生作业帮助网 编辑:作业帮 时间:2024/04/20 02:15:50
ΔABC a^2=b(b+c)证A=2B

ΔABC a^2=b(b+c)证A=2B
ΔABC a^2=b(b+c)证A=2B

ΔABC a^2=b(b+c)证A=2B
a²=b(b+c)=b²+bc,则a²-b²=bc;
cosB=(a²+c²-b²)/(2ac)=(bc+c²)/(2ac)=(b+c)/(2a);
要求证A=2B,则a/sinA=b/sinB,a/sin(2B)=b/sinB,
a/(2sinBcosB)=b/sinB,则cosB=a/(2b);
则(b+c)/(2a)=a/(2b),变形,得a²=b(b+c);
逆推回去,得证!
用余弦定理:
cosA = ( b^+c^-a^ )/ 2bc = (c-b)/2b
cosB = ( a^+c^-b^ )/ 2ac = (b+c)/2a
cos2B = 2(cosB)^-1 = (c-b)/2b
也即:cosA = cos2B
又 a^=b^+ bc ,所以a大于b 2B小于180度
A和2B都小于180度,只能是A=2B