Questions tagged [triangles]

The tag has no usage guidance.

25 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
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 $$\...
Henry's user avatar
  • 830
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^...
Oliver Nash's user avatar
  • 1,404
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 ...
Spiridon Kuruklis's user avatar
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 ...
Squala's user avatar
  • 964
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 ...
Nandakumar R's user avatar
  • 5,473
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)....
Noah Schweber's user avatar
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 ...
Đào Thanh Oai's user avatar
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 ...
Stabilo's user avatar
  • 1,479
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, ...
tisydi's user avatar
  • 335
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 ...
Matthieu Latapy's user avatar
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 ...
Đào Thanh Oai's user avatar
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 ...
Nandakumar R's user avatar
  • 5,473
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 $BC$, $AC$...
Shahrooz's user avatar
  • 4,746
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 ...
Nandakumar R's user avatar
  • 5,473
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 ...
Beni Bogosel's user avatar
  • 2,102
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 ...
Pedja's user avatar
  • 2,673
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 ...
Nandakumar R's user avatar
  • 5,473
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 ...
Nandakumar R's user avatar
  • 5,473
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 ...
FD_bfa's user avatar
  • 147
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-...
Nandakumar R's user avatar
  • 5,473
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 ...
Nandakumar R's user avatar
  • 5,473
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 ...
A.Zakharov's user avatar
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 $(A)$, $(B)$, $(...
Đào Thanh Oai's user avatar
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 ...
Đào Thanh Oai's user avatar
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. ...
Pierre Humbert Leblanc's user avatar