Rational numbers $a$, $b$, $c$ satisfy the equation \[(a+b+c)(a+b-c)=c^2\,.\]Show that $a+b=c=0$.
Source: 2017 Polish Junior Math Olympiad First Round
Tags: algebra, number theory
Rational numbers $a$, $b$, $c$ satisfy the equation \[(a+b+c)(a+b-c)=c^2\,.\]Show that $a+b=c=0$.