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Geometric inequality in regular pentagon
Let $ABCDE$ a regular pentagon inscribed in a circle of center $O$. Let $P$ an interior point of the pentagon from which we consider parallel line segments to all the sides of the pentagon. We know ...
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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 ...
user740859
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Prove that $|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2$. [closed]
A polygon $A_{1}A_{2}...A_{n}$ has a circumscribed circle with radius $R$. Prove $$|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2.$$
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