Questions tagged [triangles]

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A Morley type result for quadrilaterals

Consider the following sequence of known results. The bisectors of the angles of a triangle generate a degenerate triangle (i.e., one whose vertices coincide). The trisectors of the angles of a ...
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172 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 ...
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1answer
72 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
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1answer
400 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 ...
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1answer
151 views

Triangle angle bisectors, trisectors, quadrisectors,

With the triangle angle bisector theorem and Morely's trisector theorem as background, are there any pretty theorems known for triangle $n$-sectors, $n > 3$? For example, angle quadrisectors? The ...
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1answer
162 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) ...
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1answer
246 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 ...
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3answers
468 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 ...
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142 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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130 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 $(A)$, $(B)$, $(...
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192 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 ...
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213 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 ...
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2answers
473 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. ...
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2answers
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 $...
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1answer
49 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 ...
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380 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 ...
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1answer
238 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 ...
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3answers
99 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?
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99 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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100 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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135 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
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1answer
499 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 ...
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2answers
468 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 ...
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1answer
237 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 ...
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2answers
212 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}{...
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1answer
188 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 $...
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4answers
450 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 ...
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2answers
508 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 ...
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4answers
854 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 ...
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1answer
441 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 ...
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2answers
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 ...
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2answers
181 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 $r,s$ ...
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1answer
172 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 ...
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3answers
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 ...
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2answers
1k views

What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...
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3answers
773 views

Are there Heronian triangles that can be decomposed into three smaller ones?

Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...
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569 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 $$\...
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2answers
389 views

Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron

I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron. Anyone has any clue how to approach this problem?
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5answers
1k views

Impossible Heronian Triangles (Ratio of 2 Sides)

There is no Heronian triangle (or simply consider triangles on an integer lattice which also have integer side lengths) for which one side is half the length of another side. What other "side-side ...
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1answer
2k views

4-regular graphs with every edge in a triangle

I am interested in regular graphs in which every edge lies in a triangle. For 3-regular graphs, only the complete graph $K_4$ has this property, so there's not much to see here. For 4-regular graphs,...
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404 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 ...
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2answers
420 views

Triangles, squares, and discontinuous complex functions

Is there some onto function $f:$ $\mathbb{C}$ $\rightarrow$ $\mathbb{C}$ such that for each triangle $T$ (with its interior), $f(T)$ is a square (with interior, too) ? I would have the same question ...
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2answers
498 views

Routh's theorem in three dimensions

The most well known case of Routh's triangle theorem is: If the sides BC, CA,and AB are trisected at the points D, E, and F, respectively, then the area of the inside triangle formed by AD, BE, ...
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1answer
725 views

Malfatti Circles - Limiting point

"Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles" (for a brief historical account on this topic, see ...
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3answers
451 views

find the collision of a particle with a swept triangle.

Given there is triangle: V in 3D space that transforms over time t -> t1 to V1, and a static point P is somewhere in 3d space, how can I determine if P ever collides with V, and if so at what value of ...
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5answers
1k views

How to compute the average distance till intersection within a triangle in $\mathbb{R}^2$?

You are given 3 points in $\mathbb{R}^2$; $A$, $B$, $C$ forming a triangle with area > 0. You pick an arbitrary point inside $ABC$ and an arbitrary direction. After some distance $d$, you will ...
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1answer
592 views

Egalitarian measures

A question I got asked I while ago: If $T$ is a triangle in $\mathbb R^2$, is there a function $f:T\to\mathbb R$ such that the integral of $f$ over each straight segment connecting two points in the ...
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8answers
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Side-Angle-Side Congruence and the Parallel Postulate

Is there a link between the side-angle-side congruence of triangles and the parallel postulate? Specifically, does it follow from Euclid's first four axioms alone? In fact, does it even follow from ...
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1answer
1k views

How to find the Fermat Point using the construction of the tangent to ellipse?

Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point). I want a hint for solving this problem using ...
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2answers
321 views

Characterizing triangles unembeddedly

The mathedu mailing list has a recent longish thread at http://www.nabble.com/Why-do-we-do-proofs--to25809591.html which discussed among other things whether we should teach triangles as labeled or ...