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The "napkin-ring problem" sometimes shows up in 2nd-year calculus courses, but it can fit quite neatly into a high-school geometry course via Cavalieri's principle. However, the conclusion remains as …

Since you want to "explore" an idea rather than seeking an answer to a well-defined question, maybe you'd be interested in Wikipedia's article titled Villarceau circles.

Several answers point out the following: The space of irrationals is homeomorphic to the Baire space $\mathbb N^{\mathbb N}$ of functions from $\mathbb N$ to $\mathbb N$. No one gave an explicit hom …

It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then $$ a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta …

This won't answer the question, but Joseph O'Rourke got four up-votes for doing some nice graphics to clarify the question. In the case of the quadrilateral, we have $$ \sin\alpha\sin\beta\sin\gamma\ …

This item isn't getting attention, so I'll try it here: begin quote The three lines through antipodal pairs of centers of faces of a cube meet each other pairwise at $90^\circ$ angles. The three lines …

I posted this question thinking that the response would be two or three answers that say "Counterexamples to this are found in every textbook—for example this one and this one and this one." But the r …

I posted this question to stackexchange, where it's generated some comments but no progress toward answering it. I'm going to say somewhat more here than I did there. At one vertex of a pentagon ins …

Today I attended a talk by Terence Tao, attended by (I'm guessing) probably at least a couple of thousand people, in which among other things he said it had been proved that no packing of spheres in t …

I posted this question to stackexchange and after 24 hours it's got five votes and no answers, so let's see if mathoverflow can say more than that. Consider two propositions in geometry: Circumscri …

Heron's formula says the area of a triangle whose sides have lengths $a,b,c$ is $$ \frac14\sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}. $$ It is true that this is an "opaque formula" with which you "just chuck …

If you can reduce the problem to that of finding the shortest path between two points at the same longitude, then you can proceed as follows. Follow some path from your point of origin to your destin …

I think the "lifting of the points into $d+1$ dimensions" referred to in Joseph O'Rourke's answer means something like this: $(x_1,\dots,x_d) \mapsto (x_1,\dots,x_d,x_1^2+\cdots+x_d^2)$. Then the edg …

I asked [a very similar question] (convexity of images of space-filling curves) here once. Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t …

This is a reference request. I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you …