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Please help

 Apr 23, 2022
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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.

 Apr 23, 2022
edited by MaxWong  Apr 23, 2022

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