With the on-going trend, Algebraic Inequalities are almost erasing the existence of Geometrical Inequalities . Nevertheless, They are equally beautiful (if not more) , although they seem a lot harder to me. Yes, they can mostly be proved by brute-force trigononmetry or complex (whatever it is , they are equally good solutions) , but I am going to focus on something a bit different to both .

First things first – You can’t multiply, if you can’t add. –

**Triangle Inequality**

If are the side-lengths of triangle ABC , then the following is true –

Note : The equality case is when they triangle is degenerate or are collinear.

Also, we have the following important and useful observation –

*The side opposite the greatest angle is greater. *

Mathematically , If then ,

So , If we can find it three positive reals satisfying our triangle inequality, then it is safe to conclude that they are the side-lengths of a triangle.

Surprisingly, quite a few good problems can be solved by this fact alone.

**Problems **

1. If are positive numbers such that – , Then show that –

2. If are positive reals, Prove that –

3. If are side-lengths of a triangle , Prove that –

4. If are side-lengths of a triangle, Prove that –

*(Source : IMO 1983)*

**Hints and Solutions**

1. Case work , Trignonmetry

2. Cosine rule

3. Triangle Inequality –

Let be the maximum side of the triangle.

By Triangle Inequality –

Also, (*why ?) *

Hence proved.

4. By Triangle Inequality , $latexb \ge a-c$

Constructing, analogous inequalities and using them in the LHS we get –

The last inequality is Schur’s Inequality of degree 1.

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