3
votes
3answers
225 views

Points contained in a disk [on hold]

I have a question, but not sure how to prove this. We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points. Conjecture: There ...
15
votes
1answer
408 views

Is the perimeter of an ellipse with integer axes irrational?

Let $Q$ be an ellipse with integer-length axes $a$ and $b$: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$ The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$: $4 ...
22
votes
3answers
649 views

Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:       Starting with any triangle $T$, one forms $T'$ by connecting ...
5
votes
1answer
155 views

Soddy-type relation for Steiner chains

For Steiner $n$-chains of circles of radii $r_1,\dots,r_n$ tangent to an inner circle of radius $r_-$ and an outer circle of radius $r_+$, is there a Soddy-type relation between the $n+2$ quantities ...
24
votes
3answers
887 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$. ...
5
votes
3answers
945 views

Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it on a book a few years ago. Though I've tried to solve it, I'm facing difficulty. Question: You know on a plane there is an ...
5
votes
1answer
122 views

2-layer tilings with a center-of-gravity constraint

I've encountered a tiling problem with a physical constraint that might place it outside the literature on tiling. "Tiling" is a bit of a misnomer; it is a special type of cover. All the tiles are ...
5
votes
1answer
199 views

Symmetric black-hole curves

Is there a curve $C$ that connects $(0,1)$ to $(a,0)$ for some $a>0$, and, when reflected to $C'$ in the $x$-axis, the shape $S=C \cup C'$ has the property that each horizontal light ray entering ...
5
votes
1answer
287 views

Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?

The question is in the title: Let $E$ be an origin-centered ellipse in ${\mathbb R}^2$ and let $S$ be an "$L^p$-circle": $S = \{(x,y) : |x|^p + |y|^p = \text{const}\}$, where $1 \leq p \leq \infty$. ...
14
votes
3answers
968 views

Can Morley's theorem be generalized?

Morley's theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. In a talk some years ago, David Rusin made the ...
6
votes
1answer
592 views

Quadrature of the Lune

What is a good reference for the following result which I believe is proved by Tchebotarev. There are exactly 5 types of Lunes that are squarable. (Hippocrates produced three and then two more were ...
6
votes
1answer
289 views

Reorienting a ladder among $\mathbb{Z}^2$ poles

Imagine an object, which I'll call a ladder $\cal{L}$, a "racetrack" shape composed of a rectangle of length $L$ capped at either end by semicircles of radius $r$; so it is $L+2r$ tip-to-tip. View ...
6
votes
0answers
663 views

Interpolating points with minimum curvature constraint

I have $n$ points $p_i$ strictly interior to a rectangle $R$, and I would like to connect them with a curve $C$ whose curvature is as low as possible. Let $\kappa_\max(C)$ be the sharpest (largest ...
20
votes
4answers
869 views

Pinball on the infinite plane

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle ...
10
votes
1answer
444 views

Polygons uniquely inducing arrangements

A beautiful, relatively recent result is that, Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$. In a simple arrangement, every pair of lines ...
11
votes
1answer
465 views

Ratio of circumscribed/inscribed $(n{-}1)$-gons

As a discrete analog of the MO question, "Löwner-John Ellipsoid: incribed and circumscribed," I've been wondering what might be the maximum ratio of this quantity? Let $P$ be a convex polygon of $n$ ...
1
vote
5answers
1k views

Quadrilateral from 4 random points

Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points? I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q. But ...