All Questions
38 questions
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
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
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
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
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
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
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
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
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
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
2
votes
1
answer
802
views
Three circles meet at a point [closed]
I am looking for the proof of the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with circumcenter $O$. Let $A',B',C'$ be a reflection points of the points $A,B,C$ ...
- 2,661
2
votes
2
answers
163
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 ...
- 628
2
votes
2
answers
537
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 $...
- 2,661
2
votes
1
answer
155
views
Finding angle with geometric approach [closed]
I would like to solve the problem in this picture:
with just an elementary geometric approach. I already solved with trigonometry, e.g. using the Bretschneider formula, finding that the angle $ x = ...
- 29
2
votes
1
answer
184
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 ...
- 2,661
1
vote
1
answer
159
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}...
- 2,661
1
vote
2
answers
421
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?
- 113
1
vote
0
answers
96
views
Pseudo-Droz-Farny circles
I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
- 445