All Questions
Tagged with mg.metric-geometry triangles
75 questions
35
votes
17
answers
6k
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 ...
30
votes
2
answers
2k
views
Packing an upwards equilateral triangle efficiently by downwards equilateral triangles
Consider the problem of packing an upwards-pointing unit equilateral triangle "efficiently" by downwards-pointing equilateral triangles, where "efficiently" means that there is ...
- 114k
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 ...
- 20.5k
18
votes
1
answer
644
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 ...
- 47.3k
17
votes
2
answers
1k
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. ...
- 109k
16
votes
1
answer
444
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 ...
- 32.5k
16
votes
2
answers
528
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 ...
user99087
15
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 $...
- 1,776
15
votes
1
answer
17k
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. ...
- 445
12
votes
2
answers
969
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 $AC$ , ...
- 2,661
10
votes
5
answers
738
views
Dissection proof of Heron's formula?
In his recent book, Love Triangle, Matt Parker playfully complains that Heron's formula is an "opaque formula, and I feel like you just chuck in the side-lengths, turn a series of arbitrary ...
- 82.7k
10
votes
2
answers
764
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 ...
- 163
9
votes
1
answer
297
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 ...
- 20.5k
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 ...
user112109
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 ...
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 ...
- 2,661
7
votes
3
answers
400
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 ...
- 26.9k
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 ...
- 171
7
votes
2
answers
805
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 ...
- 1,044
6
votes
2
answers
433
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 ...
- 63
6
votes
2
answers
249
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 $r,s$ ...
- 93
6
votes
2
answers
215
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 ...
- 5,979
6
votes
4
answers
691
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 ...
- 151k
6
votes
1
answer
249
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 ...
user101163
6
votes
0
answers
121
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
1
answer
782
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 ...
- 667
4
votes
3
answers
1k
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
192
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 ...
- 4,757
4
votes
2
answers
320
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}{...
- 55
4
votes
1
answer
215
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 ...
- 2,661
4
votes
2
answers
575
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, ...
- 371
4
votes
2
answers
211
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}=\...
- 2,661
4
votes
1
answer
155
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.
...
- 445
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 ...
- 43
4
votes
0
answers
182
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 ...
- 5,979
4
votes
0
answers
144
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)....
- 20.5k
4
votes
0
answers
384
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 ...
- 1,776
4
votes
0
answers
269
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, ...
- 345
3
votes
1
answer
473
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 ...
- 133
3
votes
1
answer
286
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 ...
3
votes
1
answer
152
views
Triangles that can be cut into mutually congruent and non-convex polygons
It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
- 5,979
3
votes
1
answer
285
views
Name this kimberling center
The lines which connect the vertices of a triangle with the tangent points between the Spieker circle and the medial triangle are concurrent. Which kimberling center does this point correspond to?
- 1,109
3
votes
1
answer
85
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 ...
- 2,437
3
votes
1
answer
202
views
Do the heights of an acute triangle intersect at a single point (in neutral geometry)?
A well-known result of the Euclidean planimetry says that the heights of any triangle have a common point called the orthocentre of the triangle. This result is not true in neutral geometry (i.e., ...
- 41.9k
3
votes
1
answer
378
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
160
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+...
- 1,776
3
votes
1
answer
213
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) ...
- 4,829
3
votes
1
answer
507
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 ...
- 1,776
3
votes
0
answers
231
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 ...
- 1,776
3
votes
0
answers
907
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 ...
- 1,776