# Questions tagged [triangles]

The triangles tag has no usage guidance.

80
questions

32
votes

16
answers

4k
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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 ...

4
votes

1
answer

76
views

### 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
votes

1
answer

92
views

### 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.
...

11
votes

3
answers

509
views

### 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
votes

0
answers

242
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 ...

21
votes

3
answers

1k
views

### 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
votes

3
answers

387
views

### 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 ...

1
vote

0
answers

71
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 ...

4
votes

2
answers

203
views

### 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+\...

2
votes

1
answer

213
views

### 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

249
views

### 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 ...

4
votes

2
answers

179
views

### 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}=\...

3
votes

1
answer

729
views

### 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
votes

1
answer

136
views

### 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
vote

1
answer

97
views

### 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
vote

1
answer

106
views

### 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}...

7
votes

3
answers

1k
views

### 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

197
views

### 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
votes

0
answers

60
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 ...

2
votes

1
answer

154
views

### 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
votes

1
answer

141
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

1
answer

367
views

### 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
votes

2
answers

203
views

### 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
votes

4
answers

834
views

### 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 ...

11
votes

1
answer

764
views

### 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

250
views

### 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

160
views

### 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
votes

0
answers

130
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

2
answers

242
views

### Property of triangle centers

$M$ is the intersection of 3 cevians in the triangle $ABC$.
$$AB_1 = x,\quad CA_1 = y,\quad BC_1= z.$$
It can be easily proven that for both Nagel and Gergonne points the following equation is true:
$...

4
votes

0
answers

86
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 ...

2
votes

0
answers

271
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 ...

1
vote

1
answer

88
views

### Triangles with a given outer Soddy circle of the Malfatti circles

I did a JavaScript interactive picture of the Malfatti circles of a triangle. The user can drag the vertices of the triangle and the Malfatti circles are updated accordingly.
Now, I would like to ...

4
votes

1
answer

2k
views

### Distance between point inside a triangle and its vertices [closed]

How to determine the distance between an arbitrary point inside a triangle and its vertices if side lengths are given. Is there any correlation between these distances or their sum and the lengths of ...

7
votes

4
answers

569
views

### Triangle angle bisectors, trisectors, quadrisectors, …

With the triangle
angle bisector theorem
and
Morley's trisector theorem
as background ,
are there any pretty theorems known for triangle $n$-sectors,
$n > 3$?
For example, angle quadrisectors?
The ...

3
votes

1
answer

186
views

### Please identify this triangle septic

Let $ABC$ a triangle in the plane, but $D$ a point in (R3) space, such that the angles $\phi=ADB=BDC=CDA$ are equal. Let $E$ be the footpoint of $D$ in $ABC$. $E(\phi)$ describes a (irreducible) ...

2
votes

1
answer

402
views

### On 4 random points in a rectangle [closed]

Given a bounded rectangular area, I generate 4 random points. What is the probability that the fourth point lie within a triangle formed the first 3?
How would I attack this problem? The goal is to ...

7
votes

3
answers

501
views

### Two queries on triangles, the sides of which have rational lengths

Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational.
We are aware that a positive integer is called "congruent" only if it is the area of a right ...

4
votes

0
answers

200
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, ...

0
votes

0
answers

153
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 ...

1
vote

0
answers

165
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### 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 $(A)$, $(B)$, $(...

3
votes

1
answer

430
views

### An new equilateral triangle related to the Morley triangle

Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors:
Let $ABC$ be a triangle ...

3
votes

0
answers

222
views

### Are these points known? [closed]

Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively.
From my construction by GeoGebra, I found two special points as ...

17
votes

2
answers

669
views

### Why are the medians of a triangle concurrent? In absolute geometry

This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...

10
votes

2
answers

1k
views

### Do two new special points in any triangle exist?

There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.
Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $...

2
votes

1
answer

51
views

### Triangle Center from Weighted Perfect Matchings

let $\Delta$ be the triangle whose corners $A$, $B$, $C$ points in general position in Euclidean plane and, let $D$ be a fourth point inside $\Delta$.
Question:
what is known about the ...

3
votes

0
answers

656
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

1
answer

315
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 ...

3
votes

3
answers

167
views

### Cutting a square into an infinite number of triangles constrained by two rules

Can a square be cut into an infinite number of triangles so that
a) all of them are non-similar
and
b) only a finite number of them can have a common vertex?

0
votes

0
answers

105
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. ...

1
vote

0
answers

109
views

### Expected Area of Randomly Made Triangle [closed]

Say we have a piece of length one, and then we draw twice from a bin of sticks in which there are an infinite amount of sticks with lengths evenly distributed on $[0,1]$. In cases where a triangle can ...