Let be an acute triangle. Prove the following inequality

**Solution**

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# Joy of Mathematics

## Inequality in acute triangle

## Inequality in a triangle

## Inequality

## Exponential matrix

## An arccot sum

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Let be an acute triangle. Prove the following inequality

**Solution**

The solution can be found at cut -the – knot.

Given a triangle let denote the median points of the sides respectively. Prove that

where denotes the the circumradius and the inradius respectively.

*(Adil Abdullayev / RMM)*

**Solution **[Soumava Chakraborty]

We have successively

In all the above denotes the semiperimeter of the triangle. More on Gerretsen’s inequality can be found at this link .

Let satisfying . Prove that

**Solution**

Since then the numbers

could be sides of a triangle. The area of this triangle is

However , in any triangle is holds that [Weitzenböck]

(1)

where is the area of the triangle. Thus

Let . We define

It is known that this series converges. Prove that

**Solution**

We triangulise the matrix , that is where is an invertible matrix and is an upper triangular. This is possible since our matrix is over and thus its characteristic polynomial splits. Let be its eigenvalues. Then we note that is upper triangular with in its diagonal. Hence is also upper triangular with in its diagonal. Hence

However and forall . Thus and finally

Evaluate the sum

**Solution**

Well, first of all we note that

We also recall the Fourier series expansion of which is no other than

(1)

Hence