Questions tagged [triangles]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
33 votes
16 answers
5k views

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 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 ...
Noah Schweber's user avatar
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 ...
David Eppstein's user avatar
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 ...
Math.StackExchange's user avatar
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 $$\...
Henry's user avatar
  • 830
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 ...
Mariano Suárez-Álvarez's user avatar
17 votes
2 answers
916 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. ...
Fedor Petrov's user avatar
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 ...
user avatar
15 votes
1 answer
420 views

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 ...
Gro-Tsen's user avatar
  • 29.9k
14 votes
4 answers
1k 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 ...
Pedja's user avatar
  • 2,673
13 votes
1 answer
13k views

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. ...
A.Zakharov's user avatar
12 votes
2 answers
915 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 $AC$ , ...
Pedja's user avatar
  • 2,673
12 votes
1 answer
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,...
Gordon Royle's user avatar
  • 12.3k
11 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 $...
Đào Thanh Oai's user avatar
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 ...
Đào Thanh Oai's user avatar
10 votes
2 answers
730 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 ...
Tom D.'s user avatar
  • 163
9 votes
1 answer
290 views

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 ...
Noah Schweber's user avatar
8 votes
2 answers
2k 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 ...
user avatar
8 votes
8 answers
3k views

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 ...
Micah Blake McCurdy's user avatar
8 votes
4 answers
2k 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 ...
Pedja's user avatar
  • 2,673
8 votes
0 answers
158 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^...
Oliver Nash's user avatar
  • 1,404
7 votes
3 answers
541 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 ...
R. Nandakumar's user avatar
7 votes
3 answers
398 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 ...
Stanley Yao Xiao's user avatar
7 votes
5 answers
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 ...
user2814's user avatar
  • 171
6 votes
2 answers
430 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 ...
Ivan K.'s user avatar
  • 63
6 votes
3 answers
1k 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? ...
Wolfgang's user avatar
  • 13.2k
6 votes
2 answers
216 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 $r,s$ ...
G. Flowers's user avatar
6 votes
2 answers
208 views

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 ...
Nandakumar R's user avatar
  • 5,473
6 votes
4 answers
666 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 ...
Joseph O'Rourke's user avatar
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 ...
Oai Thanh Đào's user avatar
6 votes
1 answer
248 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 ...
user avatar
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 ...
Spiridon Kuruklis's user avatar
5 votes
5 answers
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 ...
bobuhito's user avatar
  • 1,537
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 ...
jcewncjewkjcke's user avatar
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 ...
Anand's user avatar
  • 53
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+\...
Henri Cohen's user avatar
  • 11.5k
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 ...
Wiley's user avatar
  • 647
5 votes
1 answer
204 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 $...
GMB's user avatar
  • 1,379
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 ...
Squala's user avatar
  • 964
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?
Humberto José Bortolossi's user avatar
4 votes
1 answer
185 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 ...
mathlove's user avatar
  • 4,727
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 $x+2y\neq 0\neq y+2x$. ...
RavenclawPrefect's user avatar
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}{...
Ritesh Ahuja's user avatar
4 votes
1 answer
211 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 ...
Pedja's user avatar
  • 2,673
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, ...
Mark B Villarino's user avatar
4 votes
2 answers
207 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}=\...
Pedja's user avatar
  • 2,673
4 votes
1 answer
141 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. ...
A.Zakharov's user avatar
4 votes
1 answer
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 ...
Vasile Moșoi's user avatar
4 votes
0 answers
174 views

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 ...
Nandakumar R's user avatar
  • 5,473
4 votes
0 answers
143 views

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)....
Noah Schweber's user avatar