a×a×a×a b×b×b×b c×c×c×c d×d×d×d=4×a×b×c×d,以a,b,c,d为边的四边形是菱形吗？
a×a×a×a+ b×b×b×b +c×c×c×c+ d×d×d×d=4×a×b×c×d,以a,b,c,d为边的四边形是菱形吗
证明:
∵a＾4+b＾4+c＾4+d＾4=4abcd
∴2a＾4+2b＾4+2c＾4+2d＾4-8abcd=0
a＾4-2(ab＾)+b＾4+b＾4-2(bc)＾+c＾4+c＾4-2(cd)＾+d＾4
+a＾4-2(ad)＾+d＾4+2(ab)＾-4abcd+2(cd)＾+2(bc)＾-4abcd+2(ad)=0
(a＾-b＾)＾+(b＾-c＾)＾+(c＾-d＾)＾+(a＾-d＾)＾+
2(ab-cd)＾+2(bc-ad)＾=0
∴a＾-b＾=0 b＾-c＾...全部
a×a×a×a+ b×b×b×b +c×c×c×c+ d×d×d×d=4×a×b×c×d,以a,b,c,d为边的四边形是菱形吗
证明:
∵a＾4+b＾4+c＾4+d＾4=4abcd
∴2a＾4+2b＾4+2c＾4+2d＾4-8abcd=0
a＾4-2(ab＾)+b＾4+b＾4-2(bc)＾+c＾4+c＾4-2(cd)＾+d＾4
+a＾4-2(ad)＾+d＾4+2(ab)＾-4abcd+2(cd)＾+2(bc)＾-4abcd+2(ad)=0
(a＾-b＾)＾+(b＾-c＾)＾+(c＾-d＾)＾+(a＾-d＾)＾+
2(ab-cd)＾+2(bc-ad)＾=0
∴a＾-b＾=0 b＾-c＾=0 c＾-d＾=0 a＾-d＾=0
ab-cd=0 bc-ad=0
∵a＞0 b＞0 c＞0 d＞0
∴a=b=c=d
∴以a,b,c,d为边的四边形是菱形。
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