Questions tagged [triangles]
The triangles tag has no usage guidance.
100
questions
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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 ...
25
votes
6
answers
2k
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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 ...
22
votes
4
answers
987
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 ...
20
votes
1
answer
483
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 ...
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 $$\...
18
votes
1
answer
637
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 ...
17
votes
2
answers
916
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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. ...
16
votes
2
answers
514
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 ...
15
votes
1
answer
420
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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 ...
14
votes
4
answers
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 ...
13
votes
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. ...
12
votes
2
answers
915
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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$ , ...
12
votes
1
answer
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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,...
11
votes
2
answers
1k
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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 $...
11
votes
3
answers
657
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 ...
10
votes
2
answers
730
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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 ...
9
votes
1
answer
290
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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 ...
8
votes
2
answers
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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 ...
8
votes
8
answers
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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 ...
8
votes
4
answers
2k
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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 ...
8
votes
0
answers
158
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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^...
7
votes
3
answers
541
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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 ...
7
votes
3
answers
398
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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 ...
7
votes
5
answers
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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 ...
6
votes
2
answers
430
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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 ...
6
votes
3
answers
1k
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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? ...
6
votes
2
answers
216
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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 $r,s$ ...
6
votes
2
answers
208
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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 ...
6
votes
4
answers
666
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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 ...
6
votes
1
answer
709
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 ...
6
votes
1
answer
248
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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 ...
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 ...
5
votes
5
answers
1k
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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 ...
5
votes
1
answer
3k
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
3k
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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 ...
5
votes
2
answers
386
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+\...
5
votes
1
answer
776
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 ...
5
votes
1
answer
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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 $...
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 ...
4
votes
3
answers
922
views
Is there a pyramid with all four faces being right triangles? [closed]
If such a pyramid exists, could someone provide the coordinates of its vertices?
4
votes
1
answer
185
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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 ...
4
votes
1
answer
142
views
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 $x+2y\neq 0\neq y+2x$. ...
4
votes
2
answers
311
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}{...
4
votes
1
answer
211
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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 ...
4
votes
2
answers
558
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, ...
4
votes
2
answers
207
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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}=\...
4
votes
1
answer
141
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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.
...
4
votes
1
answer
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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 ...
4
votes
0
answers
174
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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
votes
0
answers
143
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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)....