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
501 questions
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Incenter-of-mass of a polygon
"Circumcenter of mass"
is a natural generalization of circumcenter to non-cyclic polygons.
CCM(P) can be defined as the weighted average of the circumcenters
of the triangles in any ...
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Simple closed curves in a simply connected domain
Let $U$ be a bounded simply connected domain in the plane. Let $K$ be the boundary (or frontier) of $U$. For every $\varepsilon>0$ is there a simple closed curve $S\subset U$ such that the ...
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Generalization of some plane geometry theorems
Conjecture: Let $A_1, A_2,\dotsc,A_n$; $B_1, B_2,\dotsc,B_n$ and $C_1, C_2,\dotsc,C_n$ be $3n$ points in the plane such that $\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=\frac{2\...
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Convex polygon shadows: Shortest equivalent segments
Let $P$ be a convex polygon.
Q1. What is the shortest collection of line segments $S$ inside $P$
with the property that both $P$ and $S$ have the same sequence of orthogonal shadows
as $P$ and $S$ ...
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Inequality in a triangle associated with Golden ratio
Let $ABC$ be arbitrary triangle, $D$, $E$, $F$ are the midpoints of $BC$, $CA$, $AB$ respectively. Define points, segments in the figure below. I am looking for a proof that:
$$DE+EF+FD \le (DG+DH+EI+...
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Weird geometry problem I found
I found this problem in a section of an old notebook, where I used to write down weird problems I came across and that I didn't know how to solve. Long story short, I rediscovered this notebook a week ...
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Random quads visible from a random point
Although the MO question Limit of lights in rooms was quickly closed,
it suggests a related question:
Q0. What is the probability that a random quadrilateral $Q$
is entirely illuminated from a ...
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Generalization of notion of convexity
I am searching for the correct term for the following, if it exists.
A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x_1, x_2\in X$ such that there exists an arc of radius $...
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Planar linkage that traces a circle from its exterior?
Q.
Is there a linkage in the plane that traces out a circle $C$
in such a manner that the interior of the disk bounded
by $C$ is never intersected by any link througout the motion?
What I mean ...
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A generalization of Barrow's inequality
More than seven years ago. I posted this problem in stackexchange:
Let $ABC$ be a triangle, $P$ be arbitrary point inside of $ABC$. Let $A_1B_1C_1$ be the tangential traingle of $ABC$. Let $A'$, $B'$,...
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Minimize total area bounded by $N$ lines in general position
Suppose we have $N$ lines in general position (any two lines, but no three lines, meet at a point) ($N\geq 3$). Let the smallest bounded region have area $1$. Determine the minimum (or possibly ...
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Brother of Japanese theorem for cyclic quadrilaterals
I am looking for a proof of a like result as follows and Higher-dimensional generalizations?
Let $A, B, C, D$ be four point with lengths of $AB, BC, CD, DA$ are $a, b, c, d$ respectively. Let $F \in ...
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Generalization of Tucker circle, Conway circle and van Lamoen circle
Theorem 9.1 in this paper as follows is a generalization of Turker circle. Turker circles is a generalization of many circles as: Cosine Circle, circum circle, First Lemoine Circle, Gallatly Circle, ...
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Please identify this triangle septic
Let $ABC$ a triangle in the plane, but $D$ a point in (R3) space, such that the angles $\phi=ADB=BDC=CDA$ are equal. Let $E$ be the footpoint of $D$ in $ABC$. $E(\phi)$ describes a (irreducible) ...
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Find the smallest circumference of a figure containing n squares [closed]
So there's a figure which contains n squares of 1 x 1, and I have to find the smallest circumference possible. I don't know if there's an algorithm behind this, I've been stuck on this for two hours ...
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First-order logic of projective planes over fields [closed]
Suppose $\mathbb{P}^2(k)$ is a projective plane defined/coordinatized over a commutative field $k$. Is the first-order logic of the plane completely determined by the first-order logic of $k$ ? (In ...
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An new equilateral triangle related to the Morley triangle
Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors:
Let $ABC$ be a triangle ...
- 1,776
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Polygons with centroid at origin and vertices on compact codimension one submanifold of $\mathbb{R}^{n}-\{0\}$
Let $M$ be a compact codimension one submanifold of $\mathbb{R}^{n}$ which does not contaion $0$ and the origin lies in the bounded component of$\mathbb{R}^{n}-\{0\}$.
Is it true to say that:
...
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What is the sequence that maximizes this distance?
I have posted this question here without answer. Maybe I can get some light here.
Suppose we are given $n$ segments $l_1,...,l_n$ in $\mathbb{R}^2$ such that $|l_i|=i,\ \forall\ i=1,...,n$, where $|...
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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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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\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
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Trigonometry and plane geometry
This will be a variation on the theme of this question, or maybe a rephrasing of it with a somewhat readjusted emphasis.
In this posting I introduced the function
\begin{align}
& f_3(\theta_1,\...
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A problem on configuration of Dao's theorem on six circumcenters
Abstract: In the figure belows: Three lines through center of pair opposite red circle are concurrent. This is a statement of Dao's theorem on six circumcenter, a new theorem in plane geometry which ...
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Cutting convex polygons into triangles of same diameter
This question continues from: Cutting convex regions into equal diameter and equal least width pieces
Definitions: The diameter of a convex region is the greatest distance between any pair of points ...
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Stewart's formula in plane geometry
In his book "Plane geometry and its groups", H. Guggenheimer proves the Stewart's formula:
If $A$, $B$, and $C$ are collinear, then for any point $Ρ$ in the plane
$$
PA^2 BC + PB^2 CA + ...
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Hypothesis: An injection from polygons into $SO(2) \times S_n$
I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...
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Continuous map from $\mathbb C^4$ to $\mathbb R$ that changes sign under circular permutation of coordinates and that is $0$ only for squares
Does there exist a continuous map $f$ from $\mathbb C^4$ to $\mathbb R$ such that:
i) there exists four distinct complex numbers $a$, $b$, $c$, $d$, s.t. $f(a,b,c,d)f(b,c,d,a)<0$
ii) for every $(...
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Are these points known? [closed]
Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively.
From my construction by GeoGebra, I found two special points as ...
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Number of distinct directions in the set $\mathbb{Z}^2 \cap B(0,R)$
We say that non-zero directions $v, w \in \mathbb{R}^2$ are equivalent if they span the same line (i.e. $\exists C \in \mathbb{R}: v = Cw$.), and distinct otherwise. Given a collection $V \subset \...
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A generalization of the Sawayama-Thebault theorem
1. Introduction
The Sawayama-Thebault theorem is one of the best nice theorem in plane geometry. The theorem has a long history. It was published in AMM in 1938 the first solution appeared in 1973 ...
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What is the probability that these four random areas can yield a tetrahedron?
This is inspired by this problem about randomly broken sticks that can form a triangle. It goes in a different direction than this generalization about randomly broken sticks that can form a ...
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A conjecture on six planes [closed]
When I read Cox's Theorem, and Clifford's Circle Theorem and Miquel six circles theorem, I found the conjecture as folowing. And I checked the conjecture by the Geogebra sofware, the conjecture is ...
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Characterizing curves that bound strictly convex regions
Consider a closed curve on the plane so that if we perform any translation and dilation on it, the resulting curve intersects the original curve at most twice. Does this property characterize the ...
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The plane cut by grids
Suppose that one has an infinite two-dimensional regular grid of spacing one. When laid on the plane it cuts it into unit squares. Now take a second (identical) grid and place it with random shift ...
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Perimeter of ellipse: Combination of two geometries
Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter of}\;\gamma)$$...
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Reference question: Poncelet theorem
A very famous theorem of Poncelet states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in ...
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Tiling with one of each shape
Q. Is there a tiling of the plane by one each of simple polygons of $n$ vertices:
one triangle, one quadrilateral, one pentagon, $\ldots$ ,
one simple polygon of $n$ vertices, $\ldots$ ?
Here a ...
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Three circles meet at a point [closed]
I am looking for the proof of the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with circumcenter $O$. Let $A',B',C'$ be a reflection points of the points $A,B,C$ ...
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Quadrilateral from 4 random points
Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points?
I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q.
But ...
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What is the minimum number of triangle centers sufficient to unambiguously describe a triangle?
I am looking for a minimal number of properties describing a triangle so that these properties are invariant to the choice of a Cartesian coordinate system as well as to the order in which the ...
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Half spaces free of roots of a given polynomial
I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version
This question is motivated by the following fact in complex variable:(I learned this fact from the book of ...
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Question on a vector inequality
Is it true that
$$
\min\left( \begin{aligned}
&\|\mathbf{u}\| + \|\mathbf{v}\| - \|\mathbf{u} + \mathbf{v}\|, \\
&\|\mathbf{u}\| + \|\mathbf{w}\| - \|\mathbf{u} + \mathbf{w}\|, \\
&\|\...
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Minimal degree of a polynomial such that $|p(z_1)| > |p(z_2)|, |p(z_3)|, ..., |p(z_n)|$
I was investigating the behavior of $p(x)^n \mod {q(x)}$, for some polynomials $p, q \in \mathbb{C}[x]$. We'll assume $q$ is squarefree. If $q(x) = (x - z_1) (x - z_2) (x - z_3) ... (x - z_n)$ for ...
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Three questions concerning lattice points on sphere surfaces
Pardon my ignorance of this topic.
Q1.
In which dimensions $d$ is it the case that, for every natural number $n$,
there exists a sphere having exactly $n$ lattice points on it $(d{-}1)$-...
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Placing Axis-parallel rectangles on 2-D plane
Can we place $n$ axis-parallel rectangles on 2D plane (e.g. four sides of each rectangle must be parallel to either x-axis or y-axis) such that for every pair of rectangles, there is a region that is ...
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fit 4 circles within a square
If I have a square and want to place four equally large circles within this square, how large can the maximum radius be (compared to the lenght of the side of the square)?
Just an answer would be ok, ...
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To cut a triangle into $n$ $p$-sided polygonal regions
Given any triangular region and two integers $n$ and $p$ which can be large and $p > 4$. It is needed to cut the triangle into $n$ $p$-gons (e.g., cut a triangle into 10 heptagons). Among the $p$-...
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139
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6-periodic billiards trajectory in acute triangle
We can construct a 3-periodic billiards trajectory in an acute triangle in a classical geometric way, say taking the altitudes. Is there a similar way to construct a 6-periodic billiards?
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Segments on a closed convex plane curve
Is it true that there does not exist a closed convex plane curve containing an infinite number of segments, belonging to distinct lines each?
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Distribution of area of randomly placed circles
I've searched the web now for ages to try and find a paper on the asymptotic distribution of the area of the union of randomly placed discs on the plane. Ideally, I would be looking for the discs to ...
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