18.43 If the four sides $a, b, c, d$ of a quadrilateral satisfy the equation $a^{4}+b^{4}+c^{4}+d^{4} = 4 a b c d$, then the quadrilateral is
(A) Any quadrilateral.
(B) Rectangle.
(C) Rhombus.
(D) Isosceles trapezoid.
(E) Trapezoid.
(Chinese Jiangsu Province Xuzhou Junior High School Mathematics Competition, 1985)
［Solution 1］Given $a^{4}+b^{4}+c^{4}+d^{4}-4 a b c d=0$, and
$$
\begin{aligned}
& a^{4}+b^{4}+c^{4}+d^{4}-4 a b c d \\
= & \left(a^{4}-2 a^{2} b^{2}+b^{4}\right)+\left(c^{4}-2 c^{2} d^{2}+d^{4}\right)+2\left(a^{2} b^{2}-2 a b c d\right. \\
& \left.+c^{2} d^{2}\right) \\
= & \left(a^{2}-b^{2}\right)^{2}+\left(c^{2}-d^{2}\right)^{2}+2(a b-c d)^{2},
\end{aligned}
$$

If the above expression is 0, then $a=b=c=d$. This means the quadrilateral is a rhombus.
Therefore, the answer is $(C)$.
［Solution 2］Given $a>0, b>0, c>0, d>0$, by the arithmetic-geometric mean inequality
$$
a^{4}+b^{4}+c^{4}+d^{4} \geqslant 4 \sqrt[4]{a^{4} b^{4} c^{4} d^{4}}=4 a b c d,
$$

Equality holds if and only if $a=b=c=d$.
Therefore, the answer is $(C)$.