3
votes
0answers
40 views

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 ...
1
vote
2answers
34 views

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 ...
1
vote
0answers
40 views

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 ...
1
vote
1answer
61 views

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 $<A_1A_2A_3\cdots A_n>$, ...
3
votes
0answers
65 views

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 ...
20
votes
9answers
4k views

What is the simplest proof of the pythagorean theorem you know? [duplicate]

Maybe enough so to explain it to children.
1
vote
2answers
219 views

Hard proof concerning the periodicity of trigonometrical functions. Is that a challenge or just trivial

i want to know if exist or if you can develop or give me ideas of a proof to show that the least number for which sine is periodic is $2\pi$ $$\neg \{\exists n\in \mathbb{R} \wedge n < 2\pi: ...
5
votes
3answers
317 views

On Ceva's Theorem?

The famous Ceva's Theorem on a triangle $\Delta \text{ABC}$ $$\frac{AJ}{JB} \cdot \frac{BI}{IC} \cdot \frac{CK}{EK} = 1$$ is usually proven using the property that the area of a triangle of ...
1
vote
0answers
227 views

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 ...
3
votes
3answers
232 views

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 ...
1
vote
0answers
92 views

Is this reading of the proposition I.4 of the Elements valid?

I have studied what Heath, Russell, and what others have said about the proposition 4 of the Book I of the Elements. So far, I understand the "problems" they see with using superposition, but I still ...
8
votes
2answers
209 views

What is it that makes this proof about rational rectangles work fundamentally?

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 ...