All Questions
Tagged with plane-geometry euclidean-geometry
104 questions
5
votes
0
answers
342
views
$N$-$th$ 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 ...
- 1,776
4
votes
2
answers
387
views
Points contained in a disk [closed]
I have a question, but not sure how to prove this.
We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points.
Conjecture: There ...
- 65
4
votes
1
answer
160
views
What curve of positive curvature minimizes distance from the origin, given length and total curvature?
Let $\textit{F}$ be the family of $C^1$ curves in $\mathbb{R}^2$ of fixed length $\bar{l}$ and fixed tangent's turning angle $\bar{k}$.
What are the curves of positive curvature in $\textit{F}$ ...
- 405
4
votes
1
answer
178
views
Coloring circles in plane
We assume that all the circles in the plane are each colored with one of two colors: red or blue.
My question 1. Does there always exist an equilateral triangle such that its circumcircle and its ...
- 705
4
votes
1
answer
215
views
Point of concurrency [closed]
I am looking for the proof of the following claim:
Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are the nine-point centers of the triangles $\triangle ...
- 2,661
4
votes
1
answer
171
views
Geometric realization of an abstract triangulation of the plane
Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy (...
- 3,376
4
votes
1
answer
1k
views
A new theorem in projective geometry
My question: I am looking for a proof of problem as following:
Introduction: When I research a theorem as following:
Theorem 1: Let $ABC$ be a triangle, let $(S)$ be a circumconic of $ABC$, let $P$...
- 1,044
4
votes
1
answer
320
views
Collinearity in bicentric polygons
Can you provide a proofs for the following two claims?
Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear.
Claim ...
- 2,661
4
votes
0
answers
384
views
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
3
votes
4
answers
513
views
Terminology for polygons
As you may know term "polygon" might mean few different things
and its meaning has to guessed from context.
By some reason I have to use few of these meaning in one place.
So I converge to the ...
- 45k
3
votes
2
answers
275
views
Four concyclic points inside bicentric quadrilateral
Can you provide a proof for the following proposition:
Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a ...
- 2,661
3
votes
1
answer
85
views
How big can a triangle be, whose sides are the perpendiculars to the sides of a triangle from the vertices of its Morley triangle?
Given any triangle $\varDelta$, the perpendiculars from the vertices of its (primary) Morley triangle to their respective (nearest) side of $\varDelta$ intersect in a triangle $\varDelta'$, which is ...
- 2,437
3
votes
1
answer
123
views
Collinearity of three significant points of bicentric pentagon
Can you provide a proof for the following claim?
Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
- 2,661
3
votes
1
answer
303
views
How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle
How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle?
See also:
Malfatti circles
- 1,776
3
votes
1
answer
145
views
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 ...
- 221
3
votes
1
answer
805
views
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 ...
- 1,776
3
votes
1
answer
418
views
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, ...
- 1,776
3
votes
1
answer
507
views
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
3
votes
1
answer
145
views
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.
...
- 1,776
3
votes
0
answers
301
views
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 ...
- 1,776
3
votes
0
answers
231
views
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 ...
- 1,776
3
votes
0
answers
906
views
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 ...
- 1,776
3
votes
0
answers
239
views
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 ...
- 1,044
2
votes
1
answer
802
views
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 $A,B,C$ ...
- 2,661
2
votes
5
answers
6k
views
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 ...
- 123
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
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
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
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
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
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
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
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
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
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
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
2
votes
1
answer
184
views
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 ...
- 2,661
1
vote
1
answer
317
views
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 ...
- 2,661
1
vote
1
answer
84
views
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 ...
- 2,661
1
vote
1
answer
385
views
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 $BE$ ...
- 1,776
1
vote
1
answer
320
views
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 $...
- 2,661
1
vote
1
answer
129
views
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 ...
- 705
1
vote
0
answers
112
views
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?
- 1,776
1
vote
0
answers
96
views
Pseudo-Droz-Farny circles
I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
- 445
1
vote
0
answers
84
views
How can construct the equilateral $A''B''C''$ such that area of $A''B''C''$ is biggest
Let $ABC$ be arbitrary triangle in a plane. Let $A'B'C'$ and $A''B''C''$ be two equilateral triangles such that $A \in B'C'$, $B \in C'A'$, $C \in A'B'$ and $A \in B''C''$, $B \in C''A''$, $C \in A''B'...
- 1,776