All Questions
10 questions
2
votes
2
answers
243
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
5
votes
1
answer
1k
views
Is this a new result about hexagon?
Let a hexagon $AB'CA'BC'$ let $AB' \cap A'B=C''$, $BC' \cap B'C = A''$, $CA' \cap C'A = B''$ then three conditions as follows equivalent:
Three lines $AA', BB', CC'$ are concurrent (let the point of ...
- 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
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
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
6
votes
0
answers
320
views
Does this plane geometry theorem have a name (well-known)?
Consider three circles $(O_1)$, $(O_2)$, $(O_3)$. Denote the homothetic center of $\{$$(O_1)$, $(O_2)$$\}$ by $A$, the homothetic center of $\{$$(O_2)$, $(O_3)$$\}$ by $B$. Let $C$, $D$ be two points ...
- 1,776
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
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
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
10
votes
0
answers
1k
views
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 ...
- 151k