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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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; $r$ ...
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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 ...
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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 ...
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Generalized figures of constant width
Is it known which plane figures $Q$ can rotate touching three given circles $A$, $B$, and $C$?
This question was asked by Lazar Lyusternik in 1946, there is only one reference to this paper that ...
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Get a point inside a polygon
I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly ...
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Visual proof of convergence for Steiner's symmetrization
I want to find a visual proof of the following fact:
For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc.
All ...
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Are two triangles with equal corresponding medians, congruent?
Is the hyperbolic or spherical analogy of the following Euclidean fact, true?
Two triangles with equal corresponding medians are congruent.
More precisely: Assume that $\Delta ABC$ and $ ...
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Thinnest covering of the plane by regular pentagons
Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...
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Smallest dilation of a quadrilateral?
What is the smallest dilation of a quadrilateral in $\mathbb{R}^d$?
This may be an open problem;
my question is: Is this indeed open?
It will take me some time to explain the terms.
The notion of ...
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Strengthened version of Isoperimetric inequality with n-polygon
Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to:
\begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...
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A chain of six circles associated with a conic
I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems:
A chain of six circles ...
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Possible new theorem in plane geometry encompassing 5 famous geometry theorems
I am looking for a proof of a generalization Napoleon theorem, Bottema theorem and Brahmagupta theorem and van Aubel theorem, and Finsler–Hadwiger theorem in one configuration, as follows:
Let four ...
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In arbitrary cyclic polygon then $\sum_{i<j} x_{ij}^\alpha \ge \sum_{i<j} y_{ij}^\alpha $
I am looking for a proof of the inequality as follows:
Conjecture: Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$. Let $B_1B_2....B_n$ be a polygon incribed the circle $(O)...
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Visibility in a growing orchard
This is a variant on Polya's orchard problem.1,2
Suppose trees are planted randomly in the plane.
The question is: How many trees are visible from the origin as
their radii grow?
More precisely, ...
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Shape rotate, intersect; repeat: disk or empty set?
This question concerns a process that iterates intersection of
randomly rotated planar shapes.
Start with a simply connected region $R_0$ in the plane,
and let $c_0$ be the centroid of $R_0$.
Rotate $...
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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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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 ...
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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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Can any sequence of consecutive integers be realized as winding numbers?
For a closed plane curve $C$, define its sequence of winding numbers to
be the sorted list of the winding numbers of each of the distinct regions
of the plane demarcated by $C$.
For example, this ...
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Are point sets of the same order type connected by continuous (order type)-preserving motion?
Given two general position point sets in $\mathbb{R}^2$ of the same size and order type, is it possible to continuously move the points of one set until they coincide with those of the other set in ...
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What's the name of this geometric mathematical modeling problem?
There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner.
I know that this is a famous problem, but what is it called?
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Inside-out polygonal dissections
A dissection of a polygon $P$
is a partition of $P$ into a finite number of pieces, which can then be rearranged
(via planar translations and rotations) and joined (without overlap) to form a new ...
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Set with small internal radius, small perimeter and prescribed area
Given a regular set $E\subset \mathbb R^2$ define
$$
R(E) = \sup\{r\colon \exists x,\ B(x,r)\subseteq E\}
$$
to be the radius of the largest circle contained in $E$ and let $|\partial E|$ be the ...
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Are angles between points enough to decide the realizability?
Let n points in the plane be given whose coordinates we don't know.
Assume, however, that for any triple of the points we know the angle.
Question: Can we decide whether the n points are realizable ...
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Curves embedding in plane
Given two closed simple(no self-intersection point) curves $C_1,C_2$ in the plane $\mathbb R^2$, is there a good way to judge whether one curve can be embedded inside the other one, here embedding ...
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Spanning trees of plane graphs containing an edge of every face
I feel sure this must be known, but can I find it??
Which connected plane graphs (graphs drawn in the plane without crossings) have a spanning tree such that at least one edge of each face is in the ...
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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 ...
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Small quadrilaterals containing a given convex region
It is easy to prove that
(*) Every convex planar set of area 1 is contained in a quadrilateral of area 2.
It is also easy to see that statement (*) remains true if the constant 2 is replaced with a ...
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Creativity and the mechanization of elementary geometry
In plane geometry, it is customary to say that checking proofs
is a mechanical process but that finding new theorems
is a creative activity.
Citing J. Hadamard, "logic only sanctions the conquests of ...
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Convergence in the Caratheodory sense and Hausdorff sense
Among Jordan domains, I understand that Caratheodory convergence is weaker than Hausdorff convergence.
But if a sequence of Jordan domains all have rectifiable boundary whose arc length are all $L$, ...
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Integer points avoiding three on a line, four on a circle
A century ago, Dudeney asked to place $16$ pawns on a chessboard with no three
on a line:
As described by David Eppstein,1 the maximum number $g_3(n)$ points that
...
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Three homothetic centers are collinear
I am looking a proof for the problem as follows:
Let a convex hexagon, such that its principal diagonals are concurrent. For each side of the hexagon, extend the adjacent sides to their ...
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Set of balls which the number of the ball intersects lines on the plane is bounded
Does there exist the set of balls(may be not disjoint) $X=\{B_i\subset\mathbb{R^2};i\in I\}$, satisfing following properties?(Note that the ball has a positive real radius)
Let the set of all lines ...
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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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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 ...
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Is $\arcsin(1/4) / \pi$ irrational?
Is $\arcsin(1/4) / \pi$ rational? An approximation given by a calculator seem to suggest that it isn't, but I found no proof. Thanks in advance!
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Does list of distances define points uniquely?
There are N points on a plane. Is it feasible to reproduce their relative location
having only the list of distances. Assuming that translation, rotation and mirror are allowed
in the result. The ...
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Two queries on triangles, the sides of which have rational lengths
Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational.
We are aware that a positive integer is called "congruent" only if it is the area of a right ...
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Maximizing the area of a region involving triangles
I thought of a question while making up an exercise sheet for high school students, and posted it on MathStackExchange but did not receive an answer (the original post is here), so I thought perhaps ...
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Computer power in plane geometry
I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
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A problem of four conics
I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof:
Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...
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Decidability of convex rearrangements of polygons
Triggered by the MO question,
"How many convex shapes can be made with the pieces of the Stomachion?," I would like to pose this question:
Q. Given $n$ polygons in a set $S$, say each with integer ...
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Closed curve whose neighborhood is as large as possible
Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this:
(ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB)
...
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Shortest curve with given convex hull
Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. ...
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To minimize the Hausdorff distance between convex polygonal regions
Definition: The Hausdorff distance is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given two convex polygonal regions P1 and P2 on the ...
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Why does the triangle groups not include a tiling by 30-30-120 triangles?
Looking at http://en.wikipedia.org/wiki/Triangle_group I begin to wonder why the definition explicitly excludes the tessellation of the Euclidean plane by 30-30-120 triangles? In terms of the ...
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Meeting a set of lines in $\mathbb{R}^n$
Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of lines in $\mathbb{R}^n$. Is there a set $M\subseteq \mathbb{R}^n$ with the following properties?
$M$ intersects all the elements of ${\...
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Continuing generalization of the Simson line
In 2014, I found a nice result in plane geometry, the result is a generalization of the Simson line theorem, and there are nine proofs for this result were published in [1]-[7]. Continuing, I find a ...
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Integral straight-line embeddings of planar graphs
Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...
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