All Questions
23 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 ...
- 20.9k
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
8
votes
0
answers
161
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^...
- 1,444
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
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
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 ...
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
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
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
1
answer
145
views
Triangle centers formed a rectangle associated with a convex cyclic quadrilateral
Similarly Japanese theorem for cyclic quadrilaterals, Napoleon theorem, Thébault's theorem, I found a result as follows and I am looking for a proof that:
Let $ABCD$ be a convex cyclic quadrilateral.
...
- 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
1
answer
202
views
The centroid, the first and second Napoleon points and $X(930)$ lie on a circle
Can you provide an elementary proof for the claim given below?
Preliminary definitions:
$X(110)=$ focus of Kiepert parabola.
$X(137)=X(110)$ of orthic triangle .
$X(930)=$ anticomplement of $X(137)$ .
...
- 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
2
answers
247
views
Six concyclic points
Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with excenters $J_A$,$J_B$ and $J_C$ . Let $G$ be the orthogonal projection of the $...
- 2,661
2
votes
1
answer
375
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 ...
- 1,776
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
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