# Inequality that is Symmetrical but not Symmetric

1) Show that for any 3 real numbers $$a, b, c$$, we have

$\sqrt{ a^2 - ab + b^2 } + \sqrt{ b^2 - bc + c^2 } \geq \sqrt{ a^2 + ac + c^2 }.$

When does equality occur?

2) If $$a, b$$ and $$c$$ are real numbers such that $$a \geq c, b \geq c, c > 0$$, show that

$\sqrt{c(a-c) } + \sqrt{ c (b-c) } \leq \sqrt{ab} .$

When does equality occur?

3) Let $$a, b$$ and $$c$$ be real numbers with absolute value less than or equal to 1. Show that

$ab+bc+ca + 1 \geq 0.$

When does equality occur?

Hint: I used a tag of #Geometric Interpretation.

Note by Calvin Lin
4 years, 1 month ago

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Diagram for (2):

http://i.imgur.com/FXhnSs5.png

- 4 years, 1 month ago

1) Angle between a and b is 60 deg, angle between b and c is 60 deg and it is a case of triangular inequality.

- 4 years, 1 month ago