# Questions tagged [triangles]

The triangles tag has no usage guidance.

102
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### Interesting geometry problem [closed]

Let n be a natural number and let M be the set of points located on the border or inside the triangle OAB , where A(n,0) and B(0,n). Find the maximum cardinality of a subset S of M with the property ...

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0
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37
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### Force–Balance Matrix for FEM [closed]

Consider 4 vertices that are grouped into 2 triangles:
I need to construct a system of equations $$ \mathbf{A} f = -f^\text{ext} $$ such that the forces $f$ balance the external forces and satisfy ...

1
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1
answer

59
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### Partitioning polygons into obtuse isosceles triangles

Ref:
Partitioning polygons into acute isosceles triangles
Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
https://math.stackexchange.com/questions/1052063/...

1
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0
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59
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### 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 ...

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0
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79
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### 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 ...

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0
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92
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### 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 ...

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93
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### Is Morley’s observation complete?

Morley’s observation states that in a triangle the intersections of trisectors proximal to a (triangle) side lie six by six on three triples of parallel lines that make angles of 60° with each other. ...

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173
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### Do the heights of an acute triangle intersect at a single point (in neutral geometry)?

A well-known result of the Euclidean planimetry says that the heights of any triangle have a common point called the orthocentre of the triangle. This result is not true in neutral geometry (i.e., ...

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1
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130
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### Finding angle with geometric approach [closed]

I would like to solve the problem in this picture:
with just an elementary geometric approach. I already solved with trigonometry, e.g. using the Bretschneider formula, finding that the angle $ x = ...

4
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163
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### 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
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1
answer

120
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### Squarefree parts of integers of the form $xy(x+2y)(y+2x)$

The motivation for this question comes from Theorem 3.3 of the 1995 paper Tilings of Triangles by M. Laczkovich, which states:
Let $x$ and $y$ be non-zero integers such that $x+2y\neq 0\neq y+2x$. ...

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59
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### Tiling with non-congruent triangles all of which have an equal angle and equal area

Reference 1: an earlier question on tiling with pair-wise non-congruent tiles: Tiling with triangles of same circumradius and inradius
Reference 2: Triangulation of polygons with all triangles having ...

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66
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### 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-...

6
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2
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199
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### Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles

Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...

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### 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 ...

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147
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### 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 ...

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141
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### 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)....

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1
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289
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### Equational theory of the orthocenter

Previously asked at MSE:
Briefly speaking, I'm looking for a description of the equational theory of the orthocenter function, $\mathsf{orth}$. By $\mathsf{orth}$ I mean the (partial) function sending ...

15
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1
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387
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### Is there a conceptual reason why so many triplets of lines in a triangle are concurrent?

One of the striking phenomena one can't help but notice in elementary Euclidean geometry is how easy it appears to be to define triples of lines in a triangle which meet in a point. Now for each ...

2
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1
answer

98
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### Triangles that can be cut into mutually congruent and non-convex polygons

It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...

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0
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### 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 ...

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16
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### Which theorems have Pythagoras' Theorem as a special case?

Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...

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answer

78
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### How big can a triangle be, whose sides are the perpendiculars to the sides of a triangle from the vertices of its Morley triangle?

Given any triangle $\varDelta$, the perpendiculars from the vertices of its (primary) Morley triangle to their respective (nearest) side of $\varDelta$ intersect in a triangle $\varDelta'$, which is ...

4
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1
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124
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### The outer Nagel points and unknown central circle

Na, Nb, Nc are the outer Nagel points. A'B'C' is the contact triangle. I claim that lines A'B', A'C', B'C' always cut the sides of the triangle NaNbNc at six points corresponding to an unknown circle.
...

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3
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617
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### An open triangle problem in plane geometry

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following:
Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...

4
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0
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327
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### 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 ...

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6
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### Are there infinitely many "generalized triangle vertices"?

Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...

7
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3
answers

395
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### Maximizing the area of a region involving triangles

I thought of a question while making up an exercise sheet for high school students, and posted it on MathStackExchange but did not receive an answer (the original post is here), so I thought perhaps ...

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0
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86
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### 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 ...

4
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2
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285
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### Vertices of hyperbolic triangle with given angles

This is probably a well-known problem in hyperbolic geometry, but here goes anyway.
In the Poincar'e upper-half plane model, I am given three angles $\alpha$, $\beta$,
and $\gamma$ with $\alpha+\beta+\...

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1
answer

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### The 4th vertex of a triangle?

I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...

3
votes

1
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283
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### Constructing an equilateral triangle using Tarski's axioms for geometry

In Euclid's first geometry proposition, he constructs an equilateral triangle given an arbitrary line segment. I was wondering if it was possible to prove this straight from Tarski's axioms for ...

3
votes

1
answer

264
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### Need help with finding all angles of 11 sided 3D object [closed]

Question: I'm an artist trying to build a hendecahedron for a project (Image below to see the shape). This object consists of 5 pentagons at the base, 1 pentagon on the bottom, then 5 quadrilaterals ...

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2
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### Six conelliptic points

Can you prove the following proposition:
Proposition. Given an arbitrary triangle $\triangle ABC$. Let $D,E,F$ be the points on the sides $AB$,$BC$ and $AC$ respectively , such that $\frac{AB}{DA}=\...

2
votes

1
answer

774
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### Three circles meet at a point [closed]

I am looking for the proof of the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with circumcenter $O$. Let $A',B',C'$ be a reflection points of the points $A,B,C$ ...

3
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1
answer

151
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### Inequality in a triangle associated with Golden ratio

Let $ABC$ be arbitrary triangle, $D$, $E$, $F$ are the midpoints of $BC$, $CA$, $AB$ respectively. Define points, segments in the figure below. I am looking for a proof that:
$$DE+EF+FD \le (DG+DH+EI+...

1
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1
answer

402
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### Line segment-triangle intersection algorithm [closed]

currently in my project I'm using signed tetrahedron volume to check whether a line segment intersects a triangle. Initially I've found this approach in the great answer provided by professor O'Rourke:...

1
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1
answer

135
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### Generalizing Bottema's theorem

Can you provide another proof for the claim given below?
Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...

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4
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1k
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### Three circles intersecting at one point

Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with nine-point center $N$ and circumcenter $O$. Let $A',B',C'$ be a reflection points ...

4
votes

1
answer

209
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### Point of concurrency [closed]

I am looking for the proof of the following claim:
Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are the nine-point centers of the triangles $\triangle ...

3
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0
answers

68
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### 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 ...

2
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1
answer

174
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### The centroid, the first and second Napoleon points and $X(930)$ lie on a circle

Can you provide an elementary proof for the claim given below?
Preliminary definitions:
$X(110)=$ focus of Kiepert parabola.
$X(137)=X(110)$ of orthic triangle .
$X(930)=$ anticomplement of $X(137)$ .
...

2
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1
answer

167
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### 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 ...

2
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2
answers

503
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### A generalization of Napoleon's theorem

Can you provide a proof for the following proposition?
Proposition. Given an arbitrary $\triangle ABC$. The $\triangle AEB$, $\triangle BFC$ and $\triangle CDA$ are constructed on the sides of the $...

2
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2
answers

231
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### Six concyclic points

Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with excenters $J_A$,$J_B$ and $J_C$ . Let $G$ be the orthogonal projection of the $...

14
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4
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1k
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### Six points on an ellipse

Can you prove the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with centroid $G$. Let $D,E,F$ be the points on the sides $AC$,$AB$ and $BC$ respectively , such ...

12
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2
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893
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### Intersection point of three circles

Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with orthocenter $H$. Let $D,E,F$ be a midpoints of the $AB$,$BC$ and $AC$ , ...

2
votes

1
answer

279
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### Expected triangle area of normal distributed vertices with colinear expectations

For the bounty the already answered problem was reformulated
This question was already answered for random variables in $\mathbb{R}^3$. Now I am looking for the solution in $\mathbb{R}^2$ that could ...

2
votes

2
answers

177
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### What is the minimum number of triangle centers sufficient to unambiguously describe a triangle?

I am looking for a minimal number of properties describing a triangle so that these properties are invariant to the choice of a Cartesian coordinate system as well as to the order in which the ...

2
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0
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135
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### 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 $BC$, $AC$...