All Questions
Tagged with plane-geometry euclidean-geometry
21 questions
8
votes
4
answers
530
views
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 ...
- 151k
14
votes
1
answer
2k
views
Dao's theorem on six circumcenters associated with a cyclic hexagon
This questions from Ngo Quang Duong's paper
In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...
- 1,044
60
votes
2
answers
4k
views
Does this geometry theorem have a name?
Start with a circle and draw two tangent circles inside. The (black) inner tangent lines to the smaller circles intersect the large circle. The (red) lines through these intersection points are ...
- 509
9
votes
2
answers
595
views
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}}\...
- 1,776
10
votes
1
answer
551
views
Formula for "cointersection" of three circles?
I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?
...
- 371
6
votes
1
answer
767
views
Using mirrors to make a non-convex polygon visible from a fixed interior point
Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
- 17.9k
6
votes
0
answers
121
views
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 ...
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
90
votes
5
answers
4k
views
Does this property characterize straight lines in the plane?
Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\...
- 4,459
34
votes
4
answers
2k
views
About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals
Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$.
...
- 4,757
32
votes
8
answers
4k
views
Can Morley's theorem be generalized?
Morley's theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.
In a talk some years ago, David Rusin made the provocative ...
- 82.7k
25
votes
6
answers
2k
views
Are there infinitely many "generalized triangle vertices"?
Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
- 21.2k
22
votes
1
answer
1k
views
Aperiodic monotile without reflections?
The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the ...
- 82.7k
18
votes
5
answers
2k
views
Definition of area
I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...
- 45k
8
votes
2
answers
1k
views
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 ...
- 575
6
votes
1
answer
311
views
Lemoine-Lozada circles
I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...
- 445
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
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
0
answers
905
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
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
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