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Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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A question on Möbius strip and Jordan curve

If $A\subset \Bbb R^2$ then is the following statement true? $\{(x,y)\in {(A\times A)/ \sim}\,\,\,|\,\, (x,y)\sim(y,x)\}\simeq$ Möbius strip $\iff A$ is a Jordan curve.
C.F.G's user avatar
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4 votes
1 answer
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Point of concurrency [closed]

I am looking for the proof of the following claim: Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are the nine-point centers of the triangles $\triangle ...
Pedja's user avatar
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4 votes
1 answer
256 views

Polar interpretation of convexity

Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...
Jennifer Gao's user avatar
4 votes
1 answer
1k views

What Islamic tiling patterns are constructible?

Eric Broug in his book Islamic Geometric Patterns gives straightedge and compass construction of some simpler patterns. It is clear his techniques will provide constructions for many Islamic patterns. ...
Brian Wichmann's user avatar
4 votes
1 answer
356 views

Left and right halves of convex curve

Let $S$ be a set of $n$ points in the plane in general position (no 3 on a line), $n$ even. A halving line is a line through $2$ points of $S$ that partitions $S$ into 2 equal parts ($(n-2)/2$ points ...
Xd00fg's user avatar
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4 votes
1 answer
239 views

Is there any Menelaus-type theorem for polynomials?

Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$. In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...
MR_BD's user avatar
  • 550
4 votes
1 answer
171 views

Geometric realization of an abstract triangulation of the plane

Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy (...
Allan Edmonds's user avatar
4 votes
1 answer
495 views

Cubic curve closest to the given set of points

Assume we are given the set $S$ of $n$ points on the real plane and want to draw a parametrized cubic curve (actually a segment of Bézier spline) with fixed startpoint in such a way, that the ...
isnmr's user avatar
  • 41
4 votes
1 answer
1k views

A new theorem in projective geometry

My question: I am looking for a proof of problem as following: Introduction: When I research a theorem as following: Theorem 1: Let $ABC$ be a triangle, let $(S)$ be a circumconic of $ABC$, let $P$...
Oai Thanh Đào's user avatar
4 votes
1 answer
101 views

Bounding the number of points at integral distance from vertices of a triangle

Can the number of points at integral distance to all three points of a non-degenerate triangle of area $A$ be bounded by $1+cA$ for some suitable constant $c$? Remark: Since it is easy to bound this ...
Roland Bacher's user avatar
4 votes
1 answer
320 views

Collinearity in bicentric polygons

Can you provide a proofs for the following two claims? Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear. Claim ...
Pedja's user avatar
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4 votes
0 answers
133 views

Curiosity about "conditional trig identities"

Perhaps this should be cross-posted on Math Stackexchange, but it came up in the context of some research mathematics (quaternion orders, etc.) In this context, I have three angles $\alpha, \beta, \...
Marty's user avatar
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4 votes
0 answers
182 views

The closest ellipse to a given triangle

Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set. Question: Given a general triangle T, to ...
Nandakumar R's user avatar
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4 votes
0 answers
384 views

Two triangles have the same centroid theorem

Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. The line through $A$ and perpendicular to $AA'$ meets the line through $B'$ and perpendicular to $BB'$ at $A_b$; The line through $A$ and ...
Đào Thanh Oai's user avatar
4 votes
0 answers
248 views

Minimal $b_2$ in Sarkisov's construction

In the paper On the structure of conic bundles. Math. USSR, Izv., 120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...
Nick L's user avatar
  • 6,995
4 votes
0 answers
269 views

Hyperbolic Intercept (Thales) Theorem

Is there an Intercept theorem (from Thales, but don't mistake it with the Thales theorem in a circle) in hyperbolic geometry? Euclidean Intercept Theorem: Let S,A,B,C,D be 5 points, such that SA, SC, ...
tisydi's user avatar
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4 votes
0 answers
164 views

Tileability and computabilty

Let $n>2$ be an integer. We consider $n$ pairs $(x_1,y_1),\dotsc,(x_n,y_n)$ in $\mathbb{N}^2$, and the polygon defined by drawing a straight line from $(x_k, y_k)$ to $(x_{k+1},y_{k+1})$ and from $(...
Dominic van der Zypen's user avatar
4 votes
0 answers
90 views

recursively convex plane curves

For the lack of a better term let's call a convex simple loop $u(t)$ recursively convex if for any $n \geq 0$ the $n$-th derivative $u^{(n)}(t)$ is a convex simple loop. We conjecture that any ...
Vadim Ogranovich's user avatar
4 votes
0 answers
352 views

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the ...
Ali Taghavi's user avatar
4 votes
0 answers
111 views

A question about complex plane algebraic curves

I would like to ask a question about plane projective curves. Let $C\subset{\mathbb P}_2={\mathbb P}(V)$ be a plane curve of degree $n\geq 3$. Then we have a non splitted exact sequence $$0\...
Hephaistos's user avatar
3 votes
2 answers
2k views

What is the name of the 65537-gon? [closed]

I know the name of the heptadecagon (17 sides) and the diacosipentacontaheptagon (257 sides). But what is the name of the polygon with 65537 sides? I am unable to figure it.
coudy's user avatar
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3 answers
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Is there a simple criterion to determine if two parallelograms intersect?

Assume we are given two parallelograms in the plane. How can I check if their intersection is nonempty? Note that I do not need to actually find the intersection.
Philipp's user avatar
  • 979
3 votes
2 answers
323 views

Minimum weight triangulation of lattice points in a circle

Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points $S$ inside or on the circle $C$ of radius $r$ centered on the origin. Let $P$ be the convex hull of $S$; so $P$ is inscribed ...
Joseph O'Rourke's user avatar
3 votes
2 answers
1k views

maximum number of shortest path among a set of n triangle obstacles

Assume that we have a two distinct points. The number of shortest path between these two points is one. When we add a triangle obstacle to the plane and this triangle intersects the line connecting ...
user23354's user avatar
3 votes
1 answer
329 views

Planar subsets with many pairs of points on distance $1$ [duplicate]

Let $X$ be a subset of $\mathbb R^2$ consisting of $n$ distinct points. Let $d_1(X)$ be the number of pairs of points of $X$ on distance $1$ from each other. Define $$d_1(n)=\sup_{X\subset \mathbb R^2|...
aglearner's user avatar
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3 votes
2 answers
284 views

Can the "Bisector" be represented by a holomorphic function?

Note: In this question, a complex number is counted as a vector initiated from the origin. ______________________________________________________________- Is there a holomorphic function $B:\...
Ali Taghavi's user avatar
3 votes
4 answers
513 views

Terminology for polygons

As you may know term "polygon" might mean few different things and its meaning has to guessed from context. By some reason I have to use few of these meaning in one place. So I converge to the ...
Anton Petrunin's user avatar
3 votes
1 answer
152 views

Triangles that can be cut into mutually congruent and non-convex polygons

It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
Nandakumar R's user avatar
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3 votes
3 answers
332 views

Cutting a square into an infinite number of triangles constrained by two rules

Can a square be cut into an infinite number of triangles so that a) all of them are non-similar and b) only a finite number of them can have a common vertex?
Shalom's user avatar
  • 513
3 votes
1 answer
672 views

Gluing Polygons

Consider all polygons whose vertices are lattice points and edges are parallel to the axes such that no more than two edges meet at a vertex. For two polygons A and B, define A+B be to the set of ...
Anonymous's user avatar
  • 413
3 votes
2 answers
275 views

Four concyclic points inside bicentric quadrilateral

Can you provide a proof for the following proposition: Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a ...
Pedja's user avatar
  • 2,661
3 votes
3 answers
565 views

Can you measure the degree of uniformity of a 2d shape?

Is there a calculation that could take the points that make of the outline of a 2 dimensional shape and provide a numeric evaluation representative of the uniformity or symmetry of the shape. Such as ...
Curious's user avatar
  • 31
3 votes
2 answers
158 views

Sub-sector covers for disks and balls

Define an open $k$-sector of a disk as the portion between two radii separated by an angle of $2\pi/k$, but open along the two radii (and closed along the circle boundary). Call a set a sub-$k$-sector ...
Joseph O'Rourke's user avatar
3 votes
1 answer
178 views

Generalizations of Directly Similar Theorem?

There is an attractive theorem that says that if two plane figures are directly similar, then so is any convex combination of them. Below, $P_1$ and $P_2$ are directly similar polygons: they have the ...
Joseph O'Rourke's user avatar
3 votes
1 answer
249 views

Space of simple polygons on $n$-vertices as a set of points in $\mathbb{R}^{2n}$

A simple polygon in $\mathbb{R}^2$ with $n$ vertices can be mapped to elements in $\mathbb{R}^{2n}$ by the list of the coordinates of its vertices. I expect there might be something interesting to ...
Chao Xu's user avatar
  • 613
3 votes
1 answer
237 views

Find the number of triangles in plane

Let $S$ be a set of $n$ points in the plane in general position. Each 3 points of S span a triangle. Total number of triangles spanned by S: $$\binom{n}{3}=\frac{n(n-1)(n-2)}{6}=\frac{1}{6} n^3-O(n^2 )...
Xd00fg's user avatar
  • 214
3 votes
2 answers
203 views

Recovering a set from its projections in varying coordinate systems - a projection hull?

Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
M.G.'s user avatar
  • 7,127
3 votes
2 answers
364 views

Equal area of sum of pair opposite polygons conjecture

I am looking for a proof that: if $A_{11}A_{12}...A_{1n}$; $A_{21}A_{22}...A_{2n}$; $\cdots$; $A_{i1}A_{i2}...A_{in}$; $\cdots$; $A_{m1}A_{m2}...A_{mn}$ are $m$ oriented regular polygons ($n$-gons), ...
Đào Thanh Oai's user avatar
3 votes
1 answer
179 views

Analytic or holomorphic extension of the ellipse perimeter function

Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$. Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^...
Ali Taghavi's user avatar
3 votes
1 answer
234 views

Large class of curves which only intersect each other finitely many times

I am trying to find a large subset of piecewise-differentiable plane curves of finite length (subsets of $\mathbb{R}^2$) with the following property: For any pair $\gamma_1, \gamma_2$ of curves in ...
Joe Previdi's user avatar
3 votes
1 answer
364 views

On distances between points on the plane

Take a set of $2n$ points in the plane and assume that no open set of diameter $1$ contains more than $n$ of these points. Question: can we pair up the points so that the distance between the points ...
TOM's user avatar
  • 2,288
3 votes
2 answers
2k views

Interpolation splines of bounded curvature

Given $n$ points $p_i=(x_i,y_i)$ on the [Euclidean] plane, and a positive real number $\rho$. Can we have a polynomial spline (e.g., natural cubic spline) passing through all these points, such that: (...
Ganesh's user avatar
  • 627
3 votes
1 answer
85 views

How big can a triangle be, whose sides are the perpendiculars to the sides of a triangle from the vertices of its Morley triangle?

Given any triangle $\varDelta$, the perpendiculars from the vertices of its (primary) Morley triangle to their respective (nearest) side of $\varDelta$ intersect in a triangle $\varDelta'$, which is ...
John Bentin's user avatar
  • 2,437
3 votes
1 answer
238 views

Least area and least perimeter triangles that contain a convex planar region - how different can they be?

Is there a planar convex region whose enclosing triangles of least perimeter and least area have different areas and different perimeters? And if so, which region maximizes the difference between the ...
Nandakumar R's user avatar
  • 5,979
3 votes
1 answer
123 views

Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim? Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
Pedja's user avatar
  • 2,661
3 votes
1 answer
190 views

On some centers of convex regions based on partitions

These questions are inspired by Yaglom and Boltyanskii's 'Convex figures'. Winternitz Theorem: If a 2D convex figure is divided into 2 parts by a line $l$ that passes through its center of gravity, ...
Nandakumar R's user avatar
  • 5,979
3 votes
1 answer
303 views

How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle

How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle? See also: Malfatti circles
Đào Thanh Oai's user avatar
3 votes
1 answer
138 views

A geometric property about certain polynomials in two variables

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$ where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...
Ali Taghavi's user avatar
3 votes
1 answer
247 views

Map from a convex polygon that increases distance

At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...
Reid Evans's user avatar
3 votes
1 answer
349 views

What is the shape of the convex $n$ -gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $B_n$? $$B_n=\sum_{1\le{i}\...
mathlove's user avatar
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