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
121 questions with no upvoted or accepted answers
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Extending a line-arrangement so that the bounded components of its complement are triangles
Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that $\{L_1,\dots,L_m\}...
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Tiling of the plane with manholes
Some shapes, such as the disk or the Releaux triangle can be used as manholes,
that is, it is a curve of constant width.
(The width between two parallel tangents to the curve are independent of the ...
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28
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Blocking light with mirrored convex objects
There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by ...
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Straight-line drawing of regular polyhedra
Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...
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Is "Escherian metamorphosis" always possible?
$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
- 21.2k
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0
answers
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The status of the journal “Forum Geometricorum”
The online journal Forum Geometricorum is a sort of central organ of elementary geometry (mainly triangle geometry and related topics). It has been published regularly since 2000 but seems to have ...
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0
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Do cut-length-minimizing equidissections exist?
Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...
- 21.2k
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Which finite sets could be packed into a square?
This question is inspired by an interesting visualization of the finite levels of von Neumann's hierarchy on Adam P. Goucher's blog, Complex Projective 4-Space.
The problem starts with a two-...
10
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0
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Boomerangs in Polya's orchard
Polya's orchard problem asks for what radius $r$ of trees
at each lattice point within a distance $R$
of the origin block all lines of sight to the exterior of the orchard.
The answer is known; $r$ ...
- 151k
10
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0
answers
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Bang's open question strengthening Tarski's planks problem
Tarski's Planks problem,
solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires
"planks" (parallel strips) of total width $\ge d$ in order to completely cover
a ...
- 151k
10
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0
answers
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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 ...
- 151k
9
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0
answers
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A new theorem (discovered in 2013) equivalent to Brianchon theorem (the old theorem) discovered in XIX century?
In 2013, I found a new problem as follows: Let six points $A_1$, $A_2$, ...$A_6$ lie on a circle $(O_1)$, and the six points $B_1$, $B_2$,...,$B_6$ lie on another circle $(O_2)$. If the quadruples $...
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9
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0
answers
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Herding sheep in a polygon
Imagine sheep fill a simple (simply connected) polygon $P$, except
at one vertex $x$ there is no sheep.
One convex vertex $g$ of $P$ is a gate through which the sheep should pass.
A herding dog sits ...
- 151k
9
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0
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A small planar set containing a large family of curves
A beautiful construction by Besicovitch and Rado [1] produces an astounding example of a compact connected plane set of measure zero containing circles of all radii $r\in(0,1]$.
A corollary to a ...
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Which subsets of the plane are similar to all their affine images?
A parabola P in the plane has the nice property that the image of P under any affine transformation is similar to P itself.
Which other subsets of the plane have this property?
I wondered aloud about ...
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What is a geometric construction corresponding to elliptic curve addition for Sharygin-isosceles triangles?
NB: this is a cross-posting from from MSE after two months with no progress (despite a bounty). It's totally elementary but I think it's cute.
Consider the elliptic curve defined by the cubic:
$$
a^...
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8
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0
answers
200
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Ricocheting pinball-like shot: Complexity?
Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...
- 151k
7
votes
0
answers
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Is the derivative the unique operation on points in the plane that preserves convexity?
Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$
Such that the ...
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Universal point sets for 1-plane graphs
It is a notorious open problem to find a smallest set of $N$ points that
permit any $n$-vertex planar graph to be drawn in the plane without
crossings, using only those $N$ points as vertices, and ...
- 151k
7
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Reversing shortest paths among unit disks
Twas the night before Christmas, and throughout M.O.
Not a question was posted, not even by Joe.
Well, let me remedy that. :-)
Let the plane contain a number of ...
- 151k
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answers
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Rigid plane curves
A curve is a continuous one-to-one image of the real line $\mathbb R$.
A space $X$ is rigid if the only homeomorphism of $X$ onto itself is the identity.
Is there a rigid curve in the plane?
I am ...
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How many equilaterals have vertices intersections of angle trisectors of a triangle?
The celebrated Morley’s theorem ensures that the interior trisectors, proximal to sides respectively, meet at vertices of an equilateral.
In the paper Trisectors like Bisectors with Equilaterals ...
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0
answers
320
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Does this plane geometry theorem have a name (well-known)?
Consider three circles $(O_1)$, $(O_2)$, $(O_3)$. Denote the homothetic center of $\{$$(O_1)$, $(O_2)$$\}$ by $A$, the homothetic center of $\{$$(O_2)$, $(O_3)$$\}$ by $B$. Let $C$, $D$ be two points ...
- 1,776
6
votes
0
answers
164
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Sets of points avoiding small angles
(1) $\mathbb{R}^2$.
I'd like to place $n$ points in the plane so that the smallest angle they
determine is as large as possible.
In a sense, such a point set is in very general position, not only
...
- 151k
6
votes
0
answers
274
views
An inequality in cyclic polygon and tangential polygon
I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following:
Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let $P$ ...
- 1,044
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0
answers
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$ n $-Cats-in-a-Bed Problem: Picking $ n $ points in a given planar domain to maximize the sum of their pairwise distances
Let $ C $ be a connected and simply connected compact subset of the plane $ \mathbb{R}^{2} $. How can we pick $ n $ points, denoted $ x_{1},\ldots,x_{n} $, lying in $ C $ such that the total sum $ \...
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answers
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Unbalanced equipartitions
Let $K$ be a compact convex set in the plane.
Say that a perimeter-halving partition of $K$
is a partition of $K$
into two pieces by a chord (a segment with endpoints
on the boundary $\partial K$) ...
- 151k
6
votes
0
answers
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Optimal planar net for catching convex shapes
Imagine you want to make a net out of string to filter and catch objects of
a certain size, minimizing the length of string employed.
(This actually arises in filtering biological impurities from ...
- 151k
6
votes
0
answers
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Minmax problem for polygons
Let $\text{Pol}_n$ be the set of all convex polygons on a plane with $n$ vertices. For $P\in \text{Pol}_n$ denote by $\text{Tr}(P)$ the set of all triangles which vertices are some vertices of $P$. I ...
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answers
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Arrangement of points, lines, and planes
Is it possible to construct a finite nontrivial arrangement of points, lines, and planes in 3-dimensional Euclidean space with the following properties?
every line is incident with four points and ...
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5
votes
0
answers
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On convex regions containing (and contained within) a given triangle
Given an arbitrary triangle T.
How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M?
Guess: for any T, ...
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answers
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Random walks in arrangements of lines in the plane
Let $\cal{A}_n$ be a simple arrangement of $n$ lines in $\mathbb{R}^2$.
(Simple: each pair of lines meet in a distinct point, i.e.,
no three lines pass through the same point.)
Start a random walk at ...
- 151k
5
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0
answers
342
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$N$-$th$ closed chain of six circles
Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...
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votes
0
answers
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Which equation of a Butterfly?
Let $A, B$ be two points and $L$ be a line on the Euclidean Plane. Take two points $J, G$ on the line $L$ such that $JG=constant$. Let $AJ$ meet $BG$ at $P$, $AG$ meet $BJ$ at $Q$, then the locus of ...
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answers
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Plane real curves such that their intersections with lines are hyperbolic
Let $R$ be an (irreducible) plane real algebraic curve (without isolated points).
Consider Zarissky closure of $R$ in ${\mathbb C}P^1$ (as real variety).
Suppose that $\lambda\in R \Rightarrow\...
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answers
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Boundary of an open, bounded and convex set in $\mathbb{R} ^n$
Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
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$L_1$ and $L_\infty$ Voronoi diagrams and tropical geometry: Connection?
I just realized that there is a visual similarity between Voronoi diagrams in
the $L_1$ and $L_\infty$ metrics (two images below)
Left: O'Rourke, "Computing Relative Neighborhood graph ...
- 151k
4
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answers
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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, \...
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answers
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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 ...
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answers
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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 ...
- 1,776
4
votes
0
answers
248
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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}...
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votes
0
answers
269
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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, ...
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4
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answers
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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 $(...
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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 ...
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A generalized ellipse
We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$
where $a,b$ are two given points in the plane and $\lambda$ is a constant.
Now we consider the ...
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0
answers
111
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A question about complex plane algebraic curves
I would like to ask a question about plane projective curves. Let $C\subset{\mathbb P}_2={\mathbb P}(V)$ be a plane curve of degree $n\geq 3$. Then we have a non splitted exact sequence
$$0\...
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votes
1
answer
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Triangle centers formed a rectangle associated with a convex cyclic quadrilateral
Similarly Japanese theorem for cyclic quadrilaterals, Napoleon theorem, Thébault's theorem, I found a result as follows and I am looking for a proof that:
Let $ABCD$ be a convex cyclic quadrilateral.
...
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3
votes
0
answers
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views
Is every weakly $1$-dimensional space embeddable in the plane?
A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
- 3,317
3
votes
0
answers
280
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Trigonometry and plane geometry
This will be a variation on the theme of this question, or maybe a rephrasing of it with a somewhat readjusted emphasis.
In this posting I introduced the function
\begin{align}
& f_3(\theta_1,\...
3
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0
answers
301
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A problem on configuration of Dao's theorem on six circumcenters
Abstract: In the figure belows: Three lines through center of pair opposite red circle are concurrent. This is a statement of Dao's theorem on six circumcenter, a new theorem in plane geometry which ...
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