# Tagged Questions

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### Area of ​​the intersection of two discs : Integral solution?

Here is the problem : We consider two cerlces that intersect in exactly two points. There $O_1$ the center of the first and $r_1$ its radius. There $O_2$ the center of the second and $r_2$ its ...
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### Geometric proof and extension of |a|=|b|=|c|=a+b+c=1 => a=1 or b=1 or c=1

We have $a,b,c\in\mathbb{C}$ verifying $|a|=|b|=|c|=a+b+c=1$, we have to show that $a=1$ or $b=1$ or $c=1$. That can be rather easily proved using trigonometry formulas. Is there a way to prove it ...
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### bounding the sum of squares of lengths of a quadrilateral inscribed in a unit square

Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is problem 1 on paper 1 of the 1989 Irish ...
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### Prove that a function defined on points in a plane is zero

Let $n\ge3$ be an integer, and $f:P\to\mathbb R$ be a function defined on any point in the plane $P$, with the property that for any regular n-gon $<A_1A_2A_3\cdots A_n>$, ...
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### Three old chestnuts in elementary geometry: is there a unified perspective? [duplicate]

Coxeter and Greitzer, in their excellent Geometry Revisited, list a few "hardy perennials" in elementary geometry: tough problems solvable by elementary methods. Their problem number 4 (on page 26 in ...
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### What is the simplest proof of the pythagorean theorem you know? [duplicate]

Maybe enough so to explain it to children.
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### What is a direct proof of isoperimetric inequality?

What is a direct proof of isoperimetric inequality? In another word, i am asking for a proof that the circle has the maximum area compare to other geometric shape with the same circumstance but ...
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### Is there a “simple” proof of the isoperimetric theorem for squares?

"Simple" means that it doesn't use any integral or multivariable calculus concepts. A friend of mine who's taking a differential calculus course came up with the problem Prove that among all the ...
I saw this problem several years ago, and I discovered a solution to it. I've since learned a somewhat more efficient solution based on the same idea. Call a rectangle in the $(x,y)$ plane ...