Question

1.三角形中,a^4+b^4+c^4=2c^2(a^2+b^2),则角C=________

Answer 1

a^4+b^4+c^4=2c^2(a^2+b^2)

===> (a^2+b^2)^2-2a^2b^2+c^4-2c^2(a^2+b^2)=0

由余弦定理得到：a^2+b^2-c^2=2abcosC

所以：a^2+b^2=c^2+2abcosC

===> (c^2+2abcosC)^2-2a^2b^2+c^4-2c^2(c^2+2abcosC)=0

===> c^4+4abc^2cosC+4a^2b^2cos^2C-2a^2b^2+c^4-2c^4-4abc^2cosC=0

===> 4a^2b^2cos^2C=2a^2b^2

===> cos^2C=1/2

===> cosC=±√2/2

===> C=45°或者135°

无法确定△ABC的形状