Questions tagged [triangles]
The triangles tag has no usage guidance.
25
questions with no upvoted or accepted answers
19
votes
0
answers
770
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 $$\...
8
votes
0
answers
158
views
What is a geometric construction corresponding to elliptic curve addition for Sharygin-isosceles triangles?
NB: this is a cross-posting from from MSE after two months with no progress (despite a bounty). It's totally elementary but I think it's cute.
Consider the elliptic curve defined by the cubic:
$$
a^...
6
votes
0
answers
110
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
149
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 ...
4
votes
0
answers
174
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 ...
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)....
4
votes
0
answers
354
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 ...
4
votes
0
answers
126
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 ...
4
votes
0
answers
251
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, ...
3
votes
0
answers
75
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 ...
3
votes
0
answers
828
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 ...
2
votes
0
answers
77
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 ...
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 $BC$, $AC$...
2
votes
0
answers
385
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 ...
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
votes
1
answer
173
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 ...
1
vote
0
answers
61
views
Cutting off odd numbers of equal area triangles from a unit square
Two earlier related posts:
Cutting the unit square into pieces with rational length sides
On a possible variant of Monsky's theorem
Question: for odd n, how does one cut the unit square into n ...
1
vote
0
answers
95
views
Tiling the plane with pair-wise non-congruent and mutually similar triangles
Question: Is it possible to tile the plane with triangles that are (1) mutually similar, (2) pairwise non-congruent and (3)non-right? No other constraints.
Note 1: Reg requirement 3 above: since any ...
1
vote
0
answers
99
views
How many convex polygons can be made from $n$ identical right angle triangles?
Whilst working on a Tangram problem, I came across the need to find the total number of convex shapes that can be produced from $16$ identical (isosceles) right angle triangles (since the Tangram can ...
1
vote
0
answers
74
views
Triangulation of polygons with all triangles having a common angle
Following Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles, we record another triangulation question.
Question: Given an n-vertex polygonal region ("n-...
1
vote
0
answers
39
views
Tiling with a one-parameter family of non-congruent triangles
This post continues Tiling with triangles of same circumradius and inradius.
The following are known about infinite sets of triangles that can be parametrized with one variable:
from an infinite set ...
1
vote
0
answers
87
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 ...
1
vote
0
answers
198
views
Some Problems On Apollonian Gasket
Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems:
Let three $(A)$, $(B)$, $(...
0
votes
0
answers
165
views
Infinity new equilateral triangles in one configuration of triangle plane
An equilateral triangle constructed from a reference triangle is a topic which is intersested by plane geometry lovers. See Napoleon equilateral triangle, Morley equilateral triangle....In this topic ...
0
votes
0
answers
115
views
Efficient Algorithm for finding area of triangle
Suppose a segment is divided into $n$ smaller segments, with each segment length determined by a breaking point chosen randomly and independently. Now take the lengths and divide them into triplets. ...