<s>[INST] equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$ In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds:
$3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$
Please prove this identity.
source: http://www.qbyte.org/puzzles/p117s.html
 [/INST] A Cayley-Menger determinant gives the volume of "tetrahedron" $PABC$:
$$288\; V^2 = \left|\begin{array}{ccccc} 
0 & 1 & 1 & 1 & 1 \\
1 & 0 & d^2 & d^2 & a^2 \\
1 & d^2 & 0 & d^2 & b^2 \\
1 & d^2 & d^2 & 0 & c^2 \\
1 & a^2 & b^2 & c^2 & 0
\end{array}\right| = d^2 \left( \left( a^2+b^2+c^2+d^2 \right)^2-3\left(a^4+b^4+c^4+d^4\right) \right)$$
Since the tetrahedron is flat, its volume is zero.
</s>