All Questions

Let us recall the statement of Pompeiu's theorem. Let $A_1A_2A_3$ be a regular triangle inscribed in a circle $\omega$. Let $X$ be an arbitrary point on the arc $A_1A_3$. Then $$|XA_1|-|XA_2|+|XA_3|=...

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question. Let us fix a regular $n$-gon with area $1$. What is the smallest ...

What axioms for geometry and trigonometry would I have to chose in order to completely avoid coordinates in defining trig functions and showing the equivalence of their geometric (unit circle) and ...

$$\frac{\pi}{4} = 2 \tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7} \;.$$ Is there some geometric construction that explains this beautiful equation (known as Hutton's formula)? Perhaps a "proof without ...