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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Intersecting balls with convex regions and a bisector thereof
This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest ...
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What is the projective dual of a planar graph?
Everybody learns the usual definition of the dual of a planar graph when edges are preserved and faces are mapped to vertices. Everybody learns the projective duality. What if we apply it to a ...
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Expected length of a certain kind of nearest-neighbor graph
Suppose I have sets of points $Z_1,\dots,Z_N$, such that $|Z_i|=m$ for all $i$, and where all $m\times N$ points are independently distributed uniformly at random in the unit square. Can someone give ...
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A relation on triplets of points in the plane
This question is a follow up of my previous one (Planar sets closed under intersection of circles, Planar sets closed under intersection of circles) and is motivated by G. Zaimi's answer https://...
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When is the area of the convex hull of a tree-like linkage maximal?
This is inspired from this recent question. Given in the plane a tree-linkage (fixed length rigid edges, vertices are flexible joints, connected and no cycles) is there a simple description of when ...
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Partitioning polygons into acute isosceles triangles
Question: Given an $N$-vertex polygon (not necessarily convex). It is to be cut into the least number of acute isosceles triangles.
Based on this MathSE discussion, one can think of a method to get $\...
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Four concyclic triangle centers
Can you prove the claim given below? Inspired by Lester's theorem I have formulated the following claim:
Claim. Given any scalene triangle $\triangle ABC$ . Let $D$ be the reflection of incenter in ...
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A rational distance problem with (possibly) multiple solutions
Background: It is not known if any point exists on the XY plane that is at rational distance from all 4 vertices of a unit square lying on the XY plane (https://mathworld.wolfram.com/...
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On convex planar regions that can be cut into only a specified number of mutually congruent and connected pieces
References:
https://math.stackexchange.com/questions/1838617/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-parts
On congruent partitions of planar regions
https://research....
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Trouble with plane embedding
Let $C$ be the middle-thirds Cantor set. Obviously $C\times [0,1]$ embeds into the plane. But $C\times D$ does not, $D$ being a closed disc in the plane.
Are there any general results which can be ...
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Number of orbits for abelian group actions
Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite.
Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...
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On a possible variant of Monsky's theorem
See Wikipedia for Monsky's theorem which states: it is not possible to dissect a square into an odd number of triangles all of equal area.
Questions: Are there quadrilaterals that allow partition into ...
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Could somebody suggest a way to determine if a parallelogram contains another parallelogram?
I thought of one way to do this.
Using the algorithm which determines if a point is inside a parallelogram,
one can determine if the polygon contains the point within $2N$ steps ($N=2$ for ...
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A generalization of Harcourt's theorem
This question is closely related to my previous question.
Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem.
Claim. Let $A_1,A_2 \ldots ...
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An asymptotic version of the Isoperimetric inequality
Let $U$ be a simply connected bounded open set in $\mathbb{R}^2$. The area of $U$ is denoted by $A$.
(We do not assume any thing about its boundary).
Assume that $\gamma_n$,s are smooth simple ...
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Settling a circular argument: room for one more?
By using a regular hexagonal arrangement it is simple to fit 19 identical circles into a larger circle of five times the radius with no circles overlapping. This leaves an area equal to six smaller ...
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A claim on the concurrency of area bisectors of planar convex regions
We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia
Definitions: Given a ...
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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 ...
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Identity map minus Cremona transformation
Let $ \delta $ be the triangle with vertices $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $\mathbb R^3$. It's a face of the standard octahedron. The Cremona transformation
$$\mathcal C: (x, y, z) \mapsto - \...
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Smallest triangles that contain 2D convex regions with reflection symmetry
Given any 2D convex region $C$ with a mirror symmetry. Two pairs of questions:
We need to find the smallest area (likewise, smallest perimeter) triangle that contains $C$. Is it sufficient to only ...
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Midpoints of three segments are collinear associated with the Pappus configuration
I am looking for proof of a problem as follows:
Let three points $A, B, C$ are collinear. Let three lines $a, a_1, a_2 $ through $A$, three lines $b, b_1, b_2 $ through $B$
three lines $c, c_1,c_2$ ...
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Maximum crossings of curvature-constrained curve
Let $C$ be a curve in the plane whose curvature is everywhere $\le 1$.
If $C$ has length $L$, what is the largest number of proper self-crossings
of $C$ as a function of $L$?
For example, the curve ...
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Homeomorphism and boundary of a complementary component
Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.
More precisely, let $h:X\to X$ be a homeomorphism.
...
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Reconstructing an ellipse from an arc, synthetically
Given only an arc of a circle, we can easily reconstruct it fully without any use of analytic geometry - indeed using only compass and straightedge. Note that by "given only an arc", we mean ...
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Cramer–Castillon problem like
Special case of Golden ratio as a property of conic section (is it known?) as follows:
Let $ABC$ be arbitrary triangle and $DEF$ is the its tangential triangle. Let $CF$ meets $AB$ at $G$ and $BE$ ...
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Isotopy Classes of Non-Connected Planar Sets
I was just looking through some of my old questions on StackExchange and noticed that this one went totally unresolved. I still think it'd be useful as a lemma for plane topology if it were true, and ...
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When is the inside of a Jordan curve open? [closed]
I'm working purely on intuition here. The Jordan curve theorem states that a Jordan curve separates the plane into a bounded component and an infinite component. For toy curves, it seems like this ...
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Enclosing a convex plane domain in a disc
The following statement seems obvious to me:
Let $\gamma:S^1\to\mathbb R^2$ be a smooth injection such that $\dot\gamma$ and $\ddot\gamma$ never vanish.
Then $\gamma$ encloses a strictly convex ...
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Vertices of Curves and Eigenvectors of Hessian
This might be a trivial question, but I can't seem to figure it out. Suppose I have an implicitly defined curve in the plane given by $f(x,y) = t$.
This curve is strictly convex, and feel free to ...
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Thirteen-point conic and four-point line, are they new?
We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...
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A formula for the area of bicentric quadrilateral
Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals.
Claim. Given bicentric quadrilateral $...
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Vertices of 2 self-polar triangles lie on conic
I have conic $\gamma$ and two self-polar triangles $ABC$, $XYZ$ with respect to my conic. Why can I construct a one conic through $ABCXYZ$?
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Coloring lines in plane
We assume that all the lines in the plane are each colored with one of two colors: red or blue. Given angle $\alpha.$
My question 1. Is there possible to get two lines with the same color and angle ...
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Relation of Convex Polygons to Halin Graphs
Can every Halin graph be visualized as the union of a planar, strictly convex polygon with its Voronoi Diagram?
It is true, that every union of a strictly convex planar polygon with its Voronoi ...
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Computational Geometric Aspects of Greedy Tour Expansion
Has the following problem already been investigated from the Computational Geometry point of view and what are the results regarding worst case complexity?
Given
a finite set $\mathcal{P}...
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Showing that a particular area is small
Note: I posted this on math.stackexchange.com earlier (original post here: https://math.stackexchange.com/questions/1471331/showing-that-a-particular-area-is-small), but it received no responses and ...
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A certain circle formed by perpendiculars
If six points are chosen, two points on each side of a triangle, such that they have the same ratio of distances to vertices, then the perpendicular lines through those points meet at six concyclic ...
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Seeking Help with Classifying Polygons: Waterholes and Airpockets in 2D Space
I am currently in the process of writing software and have encountered a mathematical problem. Perhaps there are some experts here who are familiar with this. It involves the classification of ...
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Is the formula known? and can we generalized for higher dimensions?
In this configuration as follows, we have a nice formula:
$$\cos(\varphi)=\frac{OF.OE+OB.OC}{OF.OB+OE.OC}$$
Is the formula known? and can we generalized for higher dimensions?
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The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
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On points in the interior of planar convex regions and inscribed triangles
Given any planar convex region C, it is easy to show that every point in the interior C is the mid point of at least one chord of C. Likewise,
Question: Is every point in the interior of C the ...
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A special configuration of Nine Circles Theorem and Eight Circle Theorem
The result as follows from special configuration of merge Nine Circle Theorem and Eight Circle theorem but it is new:
Problem: Let three circle $(A)$, $(B)$, $(C)$ , let $A_c$ be arbitrary point in ...
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Tiling the plane with pair-wise non-congruent and mutually similar triangles
Question: Is it possible to tile the plane with triangles that are (1) mutually similar, (2) pairwise non-congruent and (3)non-right? No other constraints.
Note 1: Reg requirement 3 above: since any ...
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Show that a region in a plane defined by a polynomial contains integer points
Let $F \in \mathbb{Z}[x,y]$ be a polynomial of degree $2n$ such that the homogeneous degree $2n$ part of $F$, say $F_{2n}$, is positive semi-definite. How does one show that for some $\delta_n > 0$ ...
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Fermat point amidst polygonal obstacles
Consider $k$ distinct points in 2D-plane with $n$ convex polygonal obstacles. Is there a poly-time algorithm (poly in $k$ and the total number of obstacle vertices) to find a point outside of all ...
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Writing the plane as {(x,y,z): x+y+z=0} [closed]
One can coordinatize the plane by choosing three axes at 120 degree angles and representing points by triples $(x,y,z)$ with $x+y+z=0$. Is there an accepted name for this kind of coordinate system? (...
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Minimum bounding rectangle of symmetric convex bodies in the plane : is the ball the worst case
The minimal rectangle containing the euclidean ball in the plane is the standard cube $B_\infty = [-1;1]^2$. I would like to know if the euclidean ball is the worst symmetric convex body to be ...
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Four incenters lie on a circle-Does this theorem have a name?
When I read the new paper 100 CHARACTERIZATIONS OF TANGENTIAL QUADRILATERALS-section 7, I remember that I posed some problem associated with tangential quadrilateral from 2014 in here and 2015 in here ...
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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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Constant width curves and inscribed/ circumscribed ellipses
It is known (see for example the Wikipedia entry on the Reuleaux triangle) that for every curve of constant width (CCW), the largest inscribed circle and the smallest circumscribed circle are ...
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