Questions tagged [triangles]
The triangles tag has no usage guidance.
29
questions with no upvoted or accepted answers
20
votes
0
answers
791
views
Series for envelope of triangle area bisectors
The lines which bisect the area of a triangle form an envelope as shown in this picture
It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is $$\...
- 840
8
votes
0
answers
160
views
What is a geometric construction corresponding to elliptic curve addition for Sharygin-isosceles triangles?
NB: this is a cross-posting from from MSE after two months with no progress (despite a bounty). It's totally elementary but I think it's cute.
Consider the elliptic curve defined by the cubic:
$$
a^...
- 1,424
6
votes
0
answers
117
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 ...
6
votes
0
answers
155
views
graphs where every cycle is a sum of triangles
I am studying a special kind of graphs, and I would like to know if they are studied in the literature and what they are called.
Let $G$ be a simple, finite, undirected, connected graph, with vertex ...
- 974
4
votes
0
answers
177
views
The closest ellipse to a given triangle
Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given a general triangle T, to ...
- 5,837
4
votes
0
answers
143
views
Is the orthocenter "(roughly) equationally finitely-based"?
Let $T$ be the "almost everywhere" equational theory of the orthocenter function, "tweaked appropriately" to avoid partiality issues (see this earlier question of mine for details)....
- 20.6k
4
votes
0
answers
366
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 ...
- 1,478
4
votes
0
answers
128
views
Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism
Let $R$ be any ring and let $A\to B\to C\to [1]$ and $A'\to B'\to C'\to [1]$ be distinguished triangles of complexes of $R$-modules. Let $f:A\to A'$, $g:B\to B'$ and $h:C\to C'$ be morphism of ...
- 1,479
4
votes
0
answers
262
views
Hyperbolic Intercept (Thales) Theorem
Is there an Intercept theorem (from Thales, but don't mistake it with the Thales theorem in a circle) in hyperbolic geometry?
Euclidean Intercept Theorem:
Let S,A,B,C,D be 5 points, such that SA, SC, ...
- 335
3
votes
0
answers
101
views
Random graphs with prescibed degrees and triangles
In short: a random graph model generates (multi-)graphs with prescribed number of edges and minimal number of triangles for each vertex. Questions arise about the actual number of triangles and the ...
- 1,434
3
votes
0
answers
856
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 ...
- 1,478
2
votes
0
answers
79
views
Another variant of the Malfatti problem
We try to add to A Variant of the Malfatti Problem
As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...
- 5,837
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 $BC$, $AC$...
- 4,746
2
votes
0
answers
393
views
On dissecting a triangle into another triangle
It is easy to see that an equilateral triangle can be cut into 2 identical 30-60-90 degrees right triangles which can then be patched together to form a 30-30-120 degrees triangle. So, via 2 ...
- 5,837
2
votes
0
answers
409
views
Important lines in triangle - reverse problem
It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there ...
- 2,172