Lebesgue Identity

Author

Eric W. Weisstein
April 13, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/LebesgueIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LebesgueIdentity.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Identity

((((a^2)+(b^2)+(c^2)+(d^2)))^2)=(((2 a d-2 b c))^2)+(((2 a c+2 b d))^2)+((((a^2)+(b^2)-(c^2)-(d^2)))^2)

Subtract@@{(a^2+b^2+c^2+d^2)^2,(a^2+b^2-c^2-d^2)^2+(2a c+2b d)^2+(2a d-2b c)^2}//Simplify

FullSimplify[(a^2+b^2-c^2-d^2)^2+(2a c+2b d)^2+(2a d-2b c)^2]