三角形abc中,若a的四次方+b的四次方+c的四次方=2c的平方(a的平方+b的平方),则角c等于

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  • (a^2+b^2-c^2)^2

    =a^4+b^4+c^4+2a^2b^2-2b^2c^2-2c^2a^2

    =2a^2b^2,

    a^2+b^2-c^2=±√2ab,

    cosC=(a^2+b^2-c^2)/2ab=±√2/2,

    C=45°或135°

    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°