# Questions tagged [polygons]

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### Explicit equation for border of the Minkowski sum of sets

Assume we have sets of the form $$M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}$$ where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$. Goal I am looking for an (explicit) representation ...
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Let $P$ and $Q$ be the boundary segments of two planar simple polygons. View these boundaries as rigid wires. Fix $Q$ in, say, the $xy$-plane, and imagine $P$ arranged in $\mathbb{R}^3$ so that $P$ ...
138 views

### Inserting points into a polygon

I want to insert $n$ points into arbitrary polygon $P$ described by ordered list of its vertexes $v_1, v_2, ..., v_m$. Each inserted point must distanced from the others on distance at least $d$. In ...
1 vote
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### Separability of graph component embeddings

I have an undirected graph. I also have an embedding of the graph in $\mathbb{R}^3$. Assume that the graph has 2 connected components. I want to know whether there exists a plane (or better - any ...
1 vote
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### To extend the Steiner-Lehmus theorem

The Steiner Lehmus theorem (https://en.wikipedia.org/wiki/Steiner%E2%80%93Lehmus_theorem) states: Every triangle with two angle bisectors of equal lengths is isosceles. Question: What could one say ...
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### The dimension of the normal cone of a face in a polytope

Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\ This seems to be intuitively obvious but I can't ...
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### Scissor congruence for foliated polygons

Given two polygons of equal area with horizontal foliations, can one describe the obstruction (if there is any but I suspect the answer to be yes) to scissor-equivalence respecting the horizontal ...
1 vote
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### Is there a non-trivial, exact analytic (symbolic) conformal map from some polygon to some rectangle?

I'm looking for any example of a conformal map $m: P \to Q$ where $P$ is some polygon of at least $4$ sides, $Q$ is a rectangle, and and $m$ is not linear (so $P$ and $Q$ are not merely scaled, ...
354 views

### Sum of the lengths of all diagonals in a regular polygon

I believe that the sum of the lengths of diagonals of a regular polygon ($n$-gon) is always greater than or equal to any other irregular polygon ($n$-gon) inscribed in a circle.. For example for a 4-...
1 vote
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### Collinearity in tangential pentagon [closed]

I am looking for a proof of the following claim: Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...
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### Happy ending problem – Why not a proof by induction?

I have been thinking for a while on the happy ending problem, looking for approaches to attack the Erdős–Szekeres conjecture: the smallest number of points for which any general position arrangement ...
1 vote
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### High-dimensional polytopes

I have two questions regarding polytopes in high dimensions, $\mathbb{R}^d$ where $d > 3$, that I could not find resources for on the web for. Suppose I have a polytope that is non-convex: How can ...
129 views

### The product of the lengths of two line segments that belong to Newton line [closed]

I am looking for the proof of the following claim: Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...
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### Need help with finding all angles of 11 sided 3D object [closed]

Question: I'm an artist trying to build a hendecahedron for a project (Image below to see the shape). This object consists of 5 pentagons at the base, 1 pentagon on the bottom, then 5 quadrilaterals ...
1 vote
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### On triangulations and "coverage" of circumcircles

Let $P$ be a convex quadrilateral defined by four vertices $a$, $b$, $c$, and $d$. Suppose that the circumcircle of $\triangle abd$ contains $c$.* Let $D(\triangle abc)$ to denote the area enclosed by ...
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### Principal diagonals of octagon meet in a single point

Can you provide a proof for the following claim: Claim. Given octagon circumscribed about an ellipse. If the vertices of the octagon lie on another ellipse then its principal diagonals meet in a ...
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### Construct by compactness (Pentagonal tiling – Rao paper)

In the (arxiv) paper, Exhaustive search of convex pentagons which tile the plane, on page 4 under the proof of Lemma 2, it is said that: "... We keep a connected component $H_d'$ of $H_{d}$ such ...
1 vote
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### Equal products of triangle areas

Can you prove the following claim: Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...
88 views

### Equal sums of line segments

I would like to see a proof of the following Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...
1 vote
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