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Prove that in any acute triangle the following inequality holds:
Since it holds that
Using Nesbitt’s inequality we see that
Equality holds if-f .
Let be a triangle. Show that
Let denote the semiperimeter of the triangle. On account of the well known relations,
Let denote the golden ratio. Prove that:
First of all we note that .
Evaluate the integral
using geometric methods.
We are working on the following figure
since the red angle is due to the triangle since ( ). Therefore , the green angle is . Finally, the area of the circular sector is equal to
where and .