equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$

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