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
2
votes
1
answer
308
views
Connecting tangents of convex curves: at some point orthogonal?
Let $a(t)$ and $b(t)$ be two smooth, nested convex curves in the plane, $t\in[0,1]$:
Suppose the parametrization of $a()$ and $b()$ is such that $\dot{a}(t)$ is ...
- 151k
2
votes
1
answer
273
views
Triangulations of point sets — obtuse and acute triangles
Given a planar configuration of points in general position. It is known that the Delaunay triangulation is the 'fattest' triangulation possible. It is also easily seen that given 7 points with 6 of ...
- 5,979
2
votes
1
answer
202
views
The centroid, the first and second Napoleon points and $X(930)$ lie on a circle
Can you provide an elementary proof for the claim given below?
Preliminary definitions:
$X(110)=$ focus of Kiepert parabola.
$X(137)=X(110)$ of orthic triangle .
$X(930)=$ anticomplement of $X(137)$ .
...
- 2,661
2
votes
2
answers
277
views
Property of triangle centers
$M$ is the intersection of 3 cevians in the triangle $ABC$.
$$AB_1 = x,\quad CA_1 = y,\quad BC_1= z.$$
It can be easily proven that for both Nagel and Gergonne points the following equation is true:
$...
- 61
2
votes
2
answers
163
views
Maximum possible number of similar three-colored triangles
I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...
- 628
2
votes
1
answer
246
views
The points of half area of a triangle
Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is ...
- 356
2
votes
1
answer
141
views
Does this result above six points follow have a name?
Does this result above six points follow have a name?
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points in the plane and $AB, CF, ED$ are concurrent and $BC, DA, FE$ are concurrent then $CD, EB, AF$ ...
- 1,776
2
votes
1
answer
138
views
On the moment of inertia of planar convex regions and possible special nature of circular disks
We consider uniform convex planar regions and lines through their center of mass and lying in the same plane as the region; each line is parametrized by an angle $\alpha$ it makes with some reference ...
- 5,979
2
votes
1
answer
256
views
What is the center of minimum distance of a region?
Suppose we have a compact plane region $R$ (not necessarily convex or connected). I am working in a problem which involves the point $p$ in $R$ that is, in average, the closest to every other point. ...
- 306
2
votes
2
answers
537
views
A generalization of Napoleon's theorem
Can you provide a proof for the following proposition?
Proposition. Given an arbitrary $\triangle ABC$. The $\triangle AEB$, $\triangle BFC$ and $\triangle CDA$ are constructed on the sides of the $...
- 2,661
2
votes
1
answer
230
views
Some inequalities on chain of circle packing
By my computation, I pose a conjecture as follows and I am looking for a proof:
Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles $(...
- 1,776
2
votes
1
answer
102
views
What are the central points of a semi-nice region in the plane?
For a convex set in Euclidean space, there is an obvious notion of its center: namely, its center of mass, which by convexity lies in the set. For a nonconvex set there is just as obviously no nice ...
- 15.4k
2
votes
1
answer
189
views
What is the maximal diameter of a cell in a particular partition of the simplex?
Consider a standard simplex with points $(p_1, \dots, p_n)$, $p_i \ge 0$, and $\sum_i p_i = 1$. Fix a set $\{q_k\}_{k=1}^K$ with $0\leq q_k \leq \infty$ and $i,j\in\{1, \dots, n\}$. Partition it via ...
- 23
2
votes
1
answer
69
views
Maximal opening angle of a polygon from a point [closed]
I'm looking for an algorithm that given a 2D convex polygon and point outside it, returns the two points of the polygon which are the two extremities of the polygon when viewed from that point.
One ...
- 121
2
votes
2
answers
183
views
A quantity associated to a triangle
Let $\Delta ABC$ be a triangle in the plane. Let $P_{1}, P_{2}, P_{3}$ be the intersection points of bisectors, medians and altitudes, respectively. We define the quantity:
\begin{equation}
Q(\Delta ...
- 356
2
votes
1
answer
399
views
An equivalence relation on the power set of the plane.
Let $R\subseteq\mathbb{R}^2$. Consider the set of all "horizontal sections" $H_R =$ {$ Rb|b\in\mathbb{R}$} where $Rb=${$ a\in\mathbb{R} | (a,b)\in R$}. Similarly consider the set of "vertical sections"...
2
votes
2
answers
242
views
A necessary and sufficient condition for three diagonals of a hexagon to be concurrent
When talking about the condition for the three diagonals of a hexagon to be concurrent, we will think of Brianchon's theorem. Using software, I discovered a necessary and sufficient trigonometric ...
- 1,776
2
votes
1
answer
107
views
Mandelbrot boundary and component of $\infty$
Let $M$ be the Mandelbrot set, and $\partial M$ its boundary. So $\partial M$ is the set of those points $z\in M$ such that every neighborhood of $z$ contains a point of $\mathbb R^2\setminus M$.
Let $...
- 3,317
2
votes
1
answer
179
views
On segments of equal area cut from planar convex regions by chords
Consider a planar convex region $C$ of unit area and all chords of it that cut off a segment of area $\alpha$ from $C$. Obviously, if $C$ is a circular disk of unit area, all segments of area $\alpha$ ...
- 5,979
2
votes
1
answer
155
views
Concyclic point made from Six arbitrary points
Let $A_1A_2A_3A_4A_5$ be irregular convex Pentagon and Let $P$ be arbitrary point anywhere in Plane geometry.
Let $X_1,X_2,X_3,X_4,X_5$ be Circumcircle of $\triangle PA1A3$; $\triangle PA2A4$; $\...
user423633
2
votes
2
answers
247
views
Six concyclic points
Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with excenters $J_A$,$J_B$ and $J_C$ . Let $G$ be the orthogonal projection of the $...
- 2,661
2
votes
1
answer
99
views
There is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon
Conjecture 1: With $n\ge 5$, given general n-polygon, there is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon (with one and only one ...
- 1,776
2
votes
1
answer
164
views
Conformal isomorphism uniquely determined by boundary identification?
Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\...
- 239
2
votes
1
answer
367
views
What is the symmetry group of this configuration?
This configuration appear as problem 3845 in Crux Mathematicorum. I see it is very beautiful. This configuration are generalization of Pascal theorem and Brianchon theorem:
Consider six points $A_1$, $...
- 1,776
2
votes
1
answer
375
views
Yiu's equilateral triangle-triplet points
In more than 2300 years since Euclid's Elements appear, there were only two equilateral triangles become famous: The Morely equilateral triangle and the Napoleon equilateral triangle. In more than ...
- 1,776
2
votes
1
answer
246
views
Even Isometries in neutral Geometry
Consider a Hilbert plane as in Hartshorne's 'Euclid and beyond' (axiomatic geometry), and its group of isometries f or 'rigid motion' generated by line reflections. Call f 'even' if it is the product ...
2
votes
1
answer
180
views
An inequality on cyclic polygon defined by Newton's identities
Let $n$-regular polygon $X_1X_2\cdots X_n$ with the circumcribed circle $(O)$. Let $n$ points $A_1, A_2,\cdots,A_n$ lie on the circle $(O)$. Let $x_{ij}=X_iX_j$ (for $1 \le i<j \le n) $. Let $a_{ij}...
- 1,044
2
votes
1
answer
253
views
A generalization of the Tucker circle theorem and the Thomsen theorem associated with a conic
I gave a generalization of the Tucker circle theorem and the Thomsen theorem at here. Now, I give a more generalization of these theorems as following:
Problem: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, ...
- 1,044
2
votes
2
answers
150
views
A plane ray which limits onto itself
A ray is a continuous one-to-one image of the half-line $[0,\infty)$.
If $f:[0,\infty)\to \mathbb R ^2$ is continuous and one-to-one, then we say that the ray $X=f[0,\infty)$ limits onto itself if for ...
- 3,317
2
votes
1
answer
84
views
What is the average component size of a coloring?
Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
- 13.4k
2
votes
1
answer
182
views
Is a line associated with antipodal points (the fact, it is the generalization of Simson line) known?
First time, I found a line associated with antipodal points, detail:
Let $ABC$ be a triangle, $(C)$ is circumconic of $ABC$. $P$ and $P'$ are two antipodal points. Construct three lines through $P'$ ...
- 1,776
2
votes
1
answer
131
views
Curves of constant width that contain triangles
Wikipedia references: Curve of constant width,
Reuleaux polygon.
We record a pair of questions on the same lines as Smallest 3-ellipses that contain triangles.
Questions:
How does one find and ...
- 5,979
2
votes
1
answer
258
views
Smallest 3-ellipses that contain triangles
Reference: https://en.wikipedia.org/wiki/N-ellipse
Question: How does one find and characterize the smallest 3-ellipses (n-ellipses with n =3) that contain a given triangle? 'Smallest' can mean 'least ...
- 5,979
2
votes
1
answer
127
views
On covering a disk by non-overlapping subdisks
I posted this question many years ago on math stackexchange but it did not get an answer. It had circulated as a puzzle in graduate school.
A disk $D$ of radius $1$ contains disks $D_i$ ($i \ge 1$) of ...
- 123
2
votes
1
answer
273
views
Checking planar convexity of 4 points with Stewart's formula
Is the following conjecture correct?
Conjecture:
If $A,B,C,D$ are four points in general position in the euclidean plane, with
$a:=\|C-B\|,\ \ b:=\|C-A\|,\ \ c:=\|B-A\|$
$a':=\|D-A\|,\...
- 13.2k
2
votes
1
answer
81
views
Could a real curve symmetric across the line be defined only by polynomial that is not reflection-invariant?
Let $R$ -- be an irreducible plane real algebraic curve (without isolated points).
Suppose that $(x,y)\in R\Leftrightarrow (x,-y)\in R.$
Question: could one find a polynomial $f(x,y)$ with zero set $...
- 413
2
votes
1
answer
1k
views
Finding integer points inside of a parallelogram
Suppose $P = \{p_1,\ldots,p_4\} \in \mathbb{R}^2$ defines a quadrilateral (here, specifically, a parallelogram). In the particular case I'm dealing with, I know that there exists at least one point ...
- 1,318
2
votes
1
answer
220
views
Maximal sets of algebraic curves, closed under rotation, dilation, and translation, that pairwise intersect at most twice
Consider a set of nontrivial algebraic curves on the plane groovy if that set is closed under rotation, dilation, and translation, and has the property that no two members of the set intersect more ...
- 785
2
votes
0
answers
114
views
Another Butterfly theorem — Conway like circle
Have You seen these result as follows before?
In Figure 1: $AA'=BB'=tAB$; $CC'=DD'=tCD$, where t is real number then $ABCD$ is a cyclic quadrilateral iff $A'B'C'D'$ is a cyclic quadrilateral.
In the ...
- 1,776
2
votes
0
answers
38
views
The trajectory of the midpoint of perimeter-bisecting pair of points of a closed convex curve and its area
Given a closed convex curve $C\subset \mathbb{R}^2$, there is a continuous family of pair of points in $C$ that bisects the perimeter of $C$. The midpoint of such pair draws a closed curve inside $C$; ...
user510735
2
votes
0
answers
213
views
A generalization of the Archimedean circle
I proposed a generalization of the Archimedean circle : In this figure $M$ is the midpoint of $AB$, $DE$; $(G)$, $(H)$, $(M)$ are the semicircles. Then two yellow circles are congruent.
Question: Is ...
- 1,776
2
votes
0
answers
119
views
Ellipse of least perimeter that contains a given triangle
This post is related to Smallest 3-ellipses that contain triangles and tries to clarify a basic issue.
Question: Given a general triangle T, How does one find and characterize the ellipse of least ...
- 5,979
2
votes
0
answers
135
views
Perimeter points in triangle
Let $ABC$ denotes a triangle and $p(ABC)$ denotes its perimeter. We say two points $O_1$ and $O_2$ inside this triangle are perimeter points if there are points $a$, $b$ and $c$ on the sides $BC$, $AC$...
- 4,784
2
votes
0
answers
70
views
Separating a certain planar region with an open set
I have a fairly specific question related to plane separation properties. I couldn't quite see how to use Phragmen–Brouwer properties to answer it because those kind of results generally apply to ...
- 517
2
votes
0
answers
53
views
How many “linkage shapes” in the plane have $n$ “joints”?
A linkage shape in the plane with $n$ joints means a choice of $n$ points (a.k.a., vertices) in the plane joined by struts (a.k.a., edges). The joints are flexible pivots for the struts connecting ...
- 303
2
votes
0
answers
182
views
Graphs determined by monohedral, edge-to-edge tilings of the plane
Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...
- 151k
2
votes
0
answers
98
views
8-partition of a planar convex body by 4 concurrent lines
It is known1 that any convex body $K$ in the plane can be
partitioned into $6$ equal-area pieces by $3$ concurrent lines
which meet at a point in $K$.
Call this a $6$-partition.
This result cannot be ...
- 151k
2
votes
0
answers
100
views
Rolling/width functions: Characterization?
Let $K$ be a strictly convex planar body of perimeter $1$.
Roll it along the $x$-axis from $0$ to $1$, and
define $f(x)$ to be the height of the highest point of $K$
when it touches at $(x,0)$.
So $f(...
- 151k
2
votes
0
answers
89
views
Irreducible real curves on ${\mathbb C}P^1$ invariant under the finite group action
Let $G$ be a finite subgroup of a Möbius group with a standart action on a real algebraic variety ${\mathbb C}{\mathbf P^1}.$
How one can describe $G$-invariant irreducible real algebraic curves?
...
- 413
2
votes
0
answers
56
views
Projecting a convex partition onto a convex set
Say that $X$ and $Y$ are two convex regions in the plane, and suppose that $X \subset Y$. Further suppose that $Y$ is partitioned into disjoint convex subsets $Y_1 ,\dots, Y_n$. Is there a way of ...
- 4,049