# Questions tagged [triangles]

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66
questions

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vote

**1**answer

30 views

### Line segment-triangle intersection algorithm

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

81 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}...

**6**

votes

**2**answers

705 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

182 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

56 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

121 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

108 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

288 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 $...

**1**

vote

**1**answer

92 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

583 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

672 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

152 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

140 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

94 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

206 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

63 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 ...

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votes

**0**answers

21 views

### Global triangular mesh parameterization for consistent tangent space

Given some triangular mesh of some arbitrary surface in $\mathbb{R}^{3}$, I am wondering how one goes about defining a consistent tangent space on the mesh. In other words, how does one construct a ...

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votes

**0**answers

237 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

78 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 ...

**3**

votes

**1**answer

1k 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 ...

**5**

votes

**3**answers

320 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

173 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

296 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 ...

**6**

votes

**3**answers

481 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

177 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, ...

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vote

**0**answers

154 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 $(A)$, $(B)$, $(...

**2**

votes

**1**answer

363 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

217 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 ...

**16**

votes

**2**answers

537 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

50 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

550 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

277 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

101 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

104 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. ...

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vote

**0**answers

102 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 ...

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votes

**0**answers

140 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

**1**answer

599 views

### Continuing generalization of the Simson line

In 2014, I found a nice result in plane geometry, the result is a generalization of the Simson line theorem, and there are nine proofs for this result were published in [1]-[7]. Continuing, I find a ...

**16**

votes

**2**answers

470 views

### Lipschitz constant for map between triangles

Let $T_1$ and $T_2$ be any two euclidean triangles with labeled sides. The sides are labeled respectively $e_1^1,e_2^1,e_3^1$ and $e_1^2,e_2^2,e_3^2$. Call $A:T_1\rightarrow T_2$ the affine map which ...

**6**

votes

**1**answer

238 views

### Problem on triangles

Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l_1,l_2,l_3$ the squares of the lengths of the sides of $T$ and $l_1^t,l_2^t,l_3^t$ the squares of the lengths of the ...

**4**

votes

**2**answers

225 views

### Inequality from a point in plane to a triangle OR Inequality on a quadrilateral

If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that :
$\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge \frac{AC}{...

**5**

votes

**1**answer

192 views

### How often can subsets of a universe intersect exactly once?

My question is inspired by the following observation:
Claim: It is not possible to choose $n$ subsets of the universe $[n]$, each of size $\Omega(n)$, such that for each subset $S$ and each element $...

**2**

votes

**4**answers

506 views

### Show that the Minkowski sum of two triangles in 3D is the union of Minkowski sums of each triangle along the other's edges?

I'd like to show (or disprove) the claim that the Minkowski sum of two triangles with vertices in $\mathbb{R}^3$, $A+B$, is equal to the union of the unions of the Minkowski sums of $A$ along all ...

**9**

votes

**2**answers

582 views

### Generalization of Stewart's theorem?

I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...

**21**

votes

**4**answers

888 views

### Triangle centers from curve shortening

The curve-shortening flow transforms curves in the plane by moving each point perpendicularly to the curve at a speed proportional to the curvature at that point. It is usually defined for smooth ...

**19**

votes

**1**answer

448 views

### Maximum height of intersection of triangles

I'd like some advice regarding the following question, which I have been struggling with for long time.
Let's call the shaded region in the below $S_3$. It is the union of three congruent isosceles ...

**2**

votes

**2**answers

159 views

### Maximum possible number of similar three-colored triangles

I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...

**6**

votes

**2**answers

188 views

### Intersecting Sets of Pythagorean Triples with Common Hypotenuses

For any $r\in\mathbb{N}$, let $A_r$ denote the set of all natural numbers that are potentially a side of a Pythagorean triple with hypotenuse $r$.
Given any $N\in\mathbb{N}$, does there exist $r,s$ ...

**4**

votes

**1**answer

177 views

### About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true?
"Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...

**5**

votes

**3**answers

3k views

### Relationship between triangle free graphs and their minimum degree

Let $T$ be a triangle-free graph on $n$ vertices with minimum degree $\delta$ (which can be $0$). How does one show that $n >2\delta -1$? It seems to be true for bipartite graphs, but I cannot see ...