# Questions tagged [geometric-inequalities]

This is a tag for geometric problems involving inequalities.

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### the sum of the reciprocals of the lengths of the interior angle bisectors of a triangle and the sum of the reciprocals of the lengths of the sides

I have been struggling with this problem: Prove that the sum of the reciprocals of the lengths of the interior angle bisectors of a triangle is greater than the sum of the reciprocals of the ...
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### In ABC find points X,Y,Z such that AXYZ is rhombus

Question - In ABC find points X,Y,Z on AB,BC,CA such that AXYZ is rhombus and area of AXYZ <= 1/2 AREA OF ABC My try - I know it very easy but I am not getting ...I take midpoints of sides ...
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### For convex cyclic hexagon $ABCDEF$, show $AC\cdot BD \cdot CE \cdot DF \cdot EA\cdot FB \geq 27\cdot AB\cdot BC\cdot CD \cdot DE\cdot EF\cdot FA$ [duplicate]

Given a convex hexagon $ABCDEF$ inscribed in the circle, prove that $$AC\cdot BD \cdot CE \cdot DF \cdot EA\cdot FB \;\geq\; 27\cdot AB\cdot BC\cdot CD \cdot DE\cdot EF\cdot FA$$ ("$AC$" means the ...
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### A point inside a triangle

I am given a triangle $\triangle ABC$ with side lengths $a,b,c$ and a point $P$ inside it. $R_A=PA$, $R_C=PC$, $R_C=PC$ the distances from point $P$ to the sides $BC, AC, AB$ are $d_a, d_b, d_c$ ...
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### If $\triangle$ is the area of triangle with side lengths $a,b,c$, then show that $\triangle \le\dfrac{1}{4}\cdot\sqrt{(a+b+c)abc}$ [duplicate]
If $\triangle$ is the area of triangle with side lengths $a,b,c$, then show that $\triangle \le\dfrac{1}{4}\cdot\sqrt{(a+b+c)abc}$. Also show that equality occurs in the above inequality when $a=b=c$ ...