+9519 We use a geometric approach because we see that if a and b are side lengths of a triangle and the included angle is 60 degrees, then the remaining side is \(\sqrt{a^2 - ab + b^2}\). Also, if the included angle is 120 degrees instead, then the remaining side is \(\sqrt{a^2 + ab + b^2}\).
(Please try to prove this using Law of Cosines.)
Refer to the following diagram:
Using triangle inequality on \(\triangle BCD\) yields the required inequalities immediately.
Exercise: Try to figure out the case when equality holds.
