All Questions
Tagged with euclidean-geometry projective-geometry
19
questions
10
votes
1
answer
506
views
A projective plane in the Euclidean plane
Problem. Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is ...
- 41.1k
3
votes
1
answer
2k
views
Does this hexagon theorem have a name?
Question : Do you know this property of a hexagon?
Consider the configuration: Six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ in a plane and let six points $B_i \in A_iA_{i+1}$ for $i=1, 2,\dots, ...
- 1,454
13
votes
2
answers
2k
views
Is it a new discovery on conic section?
I discovered a problem in plane geometry (there are some nice special cases) as follows:
Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...
- 1,454
0
votes
0
answers
92
views
Lines through the origin every pair of which meet at the same angle
This item isn't getting attention, so I'll try it here:
begin quote
The three lines through antipodal pairs of centers of faces of a cube meet each other pairwise at $90^\circ$ angles.
The three lines ...
- 12.3k
6
votes
1
answer
288
views
Does any real projective plane incidence theorem follow from axioms?
Is it known whether any projective geometry statement that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the standard projective axiomatics?
...
- 308
4
votes
1
answer
382
views
Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?
Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.
5
votes
0
answers
334
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 ...
- 1,454
2
votes
1
answer
361
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 ...
- 1,454
7
votes
0
answers
407
views
Can generalization of a generalization Pascal theorem, Pappus theorem to Higher Dimensions? [closed]
Please see a chain of six circles associated with a conic. This is a generalization of Pascal theorem, Pappus theorem. I reformulate as following:
Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a ...
- 477
16
votes
2
answers
796
views
What are Sylvester-Gallai configurations in the complex projective plane?
A Sylvester-Gallai configuration in the the complex projective plane is a finite number of $n\ge 2$ points in the complex projective plane such that there is no line through exactly two of them. ...
- 7,682
7
votes
0
answers
313
views
Status of an open question in Artin's "Geometric Algebra"
In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2).
The ...
- 501
2
votes
1
answer
244
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
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
5
votes
1
answer
428
views
Sixteen points circle - A conjecture on Möbius plane
The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note)
Consider the Bundle theorem configuration :
Points $A_1, A_2, A_3, A_4$ lie on a circle,
...
- 1,044
1
vote
1
answer
247
views
Angles and projective metric
Unless I am very wrong, the following seems to be true:
If the angle between two vectors in $\mathbb{R}^{n}_{++}$ is small, then the
value of the Hilbert projective metric between them is also ...
- 6,922