Questions tagged [triangles]
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128 questions
0 votes
1 answer
95 views
"Sphere Tetrahedron Picking" on Wolfram
(not a mathematician here) This question provides this link. And there, there are these statements: In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is ...
- 111
2 votes
0 answers
152 views
An inequality involving reciprocals of angle bisectors in a triangle [closed]
Let $ABC$ be a triangle with side lengths $a,b,c$ and internal angle bisectors $\ell_a,\ell_b,\ell_c$ corresponding to vertices $A,B,C$, respectively. I am interested in the following inequality, ...
- 29
12 votes
0 answers
334 views
Random triangle on a ring of three tangent circles: Show that the probability that the triangle contains the centre is 1/2
A triangle's vertices are random (uniform and independent) points on a ring of three mutually tangent congruent circles, with one vertex on each circle. Show that the probability that the triangle ...
- 5,049
3 votes
1 answer
255 views
A generalization of Newton–Gauss line
I am looking for a proof of a generalization of Newton–Gauss line as follows: Let ABC be a triangle, let a line $L$ meets $BC, CA, AB$ at three points $A', B', C'$ and let $A'', B'', C''$ be three ...
- 2,198
25 votes
2 answers
837 views
A tetrahedron's vertices are random points on a sphere. What is the probability that the tetrahedron's four faces are all acute triangles?
This question resisted attacks at MSE. A tetrahedron's vertices are independent uniformly random points on a sphere. What is the probability that the tetrahedron's four faces are all acute triangles? ...
- 5,049
1 vote
0 answers
132 views
Equilateral triangle centered at the centroid [closed]
I found this result a few days ago (references welcome if it's unoriginal), and am wondering if perhaps there is a simple synthetic proof for it. Let $ABC$ be a triangle with $P$ a point in the plane. ...
- 1,205
9 votes
1 answer
493 views
Decide if a pair of equal area triangles can be dissected into each other via some set of finitely many equal area pieces
Given two equal area triangles, how does one decide if both can be cut into the same set of finitely many pieces with all pieces having the same area? Does allowing the pieces to be non-convex have ...
- 7,231
4 votes
1 answer
221 views
A line of centroids
Let ABC and A'B'C' be two triangles in the plane. Let A''B''C'' be the triangle formed by the centroids of triangles AB'C, AC'B, and BA'C. Then the centroids of ABC, A'B'C', and A''B''C'' are ...
- 1,205
1 vote
0 answers
144 views
Transition to chaos in geometric iterations
Many new results have been obtained on orthic triangles iterations, so I decided to make a separate post. We are talking about orthic triangles (OT). The OT vertices are the feet of the original ...
- 568
3 votes
1 answer
496 views
Iterations of orthic triangles
The orthic triangle exists for any given triangle with the following remarks: For acute triangle orthic triangle is inscribed triangle For a rectangular triangle it is degenerated triangle (a segment)...
- 568
4 votes
1 answer
330 views
Triangles that can be cut into mutually congruent quadrilaterals
Are there triangles - other than equilateral ones - that allow partition into finitely many mutually congruent quadrilaterals? please note that we don’t allow triangles as ‘degenerate’ quads. The ...
- 7,231
0 votes
1 answer
171 views
Expressing triangle area as a product of inradius and exradii: known result?
I encountered the following relationship while exploring triangle area computations. Given a triangle with inradius $r$, exradii $r_a, r_b, r_c$, and semiperimeter $s$, the area $\Delta$ satisfies: $$ ...
11 votes
2 answers
949 views
Japanese-style geometry problem: seven circles in a triangle, show that one is congruent with another two
I made up the following Sangaku problem. In the diagram below, circles of the same color are congruent. Wherever things look tangent, they are tangent. Show that the red circle is congruent with the ...
- 5,049
24 votes
3 answers
2k views
Is there a simpler proof for this simple geometrical result? (An equilateral triangle contains three congruent circles, prove two lengths are equal.)
In equilateral $\triangle ABC$, $D$ is on $AB$, $E$ is on $AC$, and the incircles of $\triangle ADE$, $\triangle DBE$ and $\triangle EBC$ are congruent. Prove that $BD=DE$. I asked this question on ...
- 5,049
0 votes
1 answer
118 views
Conjecture on a cyclic quadrilateral associated with central line of triangle
Using computer I found a conjecture as follows (click to check by geogebra): Conjecture: Let $A$, $B$, $C$, $D$ lie on a circle such that $A$, $B$, $C$, $D$ are not formed an Isosceles trapezoid. ...
- 2,198
4 votes
3 answers
558 views
On 3 centers of triangles
We add a little to On two centers of convex regions For any interior point $P$ in a planar convex region $C$, we define d_max to be the maximum distance from $P$ to the boundary of $C$ (distance from $...
- 7,231
12 votes
5 answers
1k views
Dissection proof of Heron's formula?
In his recent book, Love Triangle, Matt Parker playfully complains that Heron's formula is an "opaque formula, and I feel like you just chuck in the side-lengths, turn a series of arbitrary ...
- 88.2k
2 votes
1 answer
378 views
Question on a vector inequality
Is it true that $$ \min\left( \begin{aligned} &\|\mathbf{u}\| + \|\mathbf{v}\| - \|\mathbf{u} + \mathbf{v}\|, \\ &\|\mathbf{u}\| + \|\mathbf{w}\| - \|\mathbf{u} + \mathbf{w}\|, \\ &\|\...
- 171
1 vote
0 answers
201 views
A certain circle formed by perpendiculars
If six points are chosen, two points on each side of a triangle, such that they have the same ratio of distances to vertices, then the perpendicular lines through those points meet at six concyclic ...
- 1,205
6 votes
2 answers
414 views
Question on a min inequality
Is it true that $$ \min\left(a^2 + b^2 - \sqrt{a^4 + b^4 + 2a^2b^2\cos(x)}, b^2 + c^2 - \sqrt{b^4 + c^4 + 2b^2c^2\cos(x-y)}, a^2 + c^2 - \sqrt{a^4 + c^4 + 2a^2c^2\cos(y)}\right) \leq \frac{1}{3} $$ ...
- 171
3 votes
1 answer
189 views
Point of concurrency of three circles which pass through vertices of a triangle and erected equilateral triangles
Let $A, B$, and $C$ be the vertices of a given triangle. Let $ACD, ABF$, and $BCE$ form equilateral triangles (internal or external). Then circles $ADF, BEF$, and $CDE$ are concurrent at point $G$. ...
- 1,205
3 votes
2 answers
290 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. ...
- 2,198
1 vote
0 answers
106 views
Name of the perspector of the orthic triangle and excentral triangle
The orthic triangle and tangential triangles of a given triangle are in perspective. What's the official kimberling center associated with this perspector?
- 1,205
3 votes
1 answer
321 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?
- 1,205
1 vote
0 answers
151 views
A circle is inscribed in a triangle, with three other circles in the corner regions. The radii are integers. Possible values of the largest radius?
Originally posted at MSE. A circle with integer radius $R$ is inscribed in a triangle. Three other circles with integer radii $a,b,c$ are each tangent to the large circle and two sides of the ...
- 5,049
1 vote
1 answer
221 views
An "almost" geodesic dome
A regular $ n$-gon is inscribed in the unit circle centered in $0$. We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
- 557
13 votes
8 answers
2k views
The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$. There is a convoluted proof that $P(ab>c)=\frac12$. But since the ...
- 5,049
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 ...
- 120k
2 votes
0 answers
122 views
Another variant of the Malfatti problem
We try to add to A Variant of the Malfatti Problem As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...
- 7,231
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?
1 vote
1 answer
210 views
Partitioning polygons into obtuse isosceles triangles
Ref: Partitioning polygons into acute isosceles triangles Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles https://math.stackexchange.com/questions/1052063/...
- 7,231
4 votes
1 answer
231 views
Cutting off odd numbers of equal area triangles from a unit square
Two earlier related posts: Cutting the unit square into pieces with rational length sides On a possible variant of Monsky's theorem Question: for odd n, how does one cut the unit square into n ...
- 7,231
4 votes
1 answer
267 views
Tiling the plane with pair-wise non-congruent and mutually similar triangles
Question: Is it possible to tile the plane with triangles that are (1) mutually similar, (2) pairwise non-congruent and (3)non-right? No other constraints. Note 1: Reg requirement 3 above: since any ...
- 7,231
1 vote
1 answer
195 views
How many convex polygons can be made from $n$ identical right angle triangles?
Whilst working on a Tangram problem, I came across the need to find the total number of convex shapes that can be produced from $16$ identical (isosceles) right angle triangles (since the Tangram can ...
- 191
3 votes
1 answer
241 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., ...
- 44.3k
2 votes
1 answer
207 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
5 votes
0 answers
299 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 ...
- 7,231
4 votes
1 answer
236 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 $x+2y\neq 0\neq y+2x$. ...
- 3,352
1 vote
0 answers
130 views
Triangulation of polygons with all triangles having a common angle
Following Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles, we record another triangulation question. Question: Given an n-vertex polygonal region ("n-...
- 7,231
5 votes
2 answers
413 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 ...
- 7,231
6 votes
0 answers
152 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 ...
8 votes
1 answer
357 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 ...
- 1,024
4 votes
0 answers
172 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)....
- 19.7k
9 votes
1 answer
370 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 ...
- 19.7k
16 votes
1 answer
508 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 ...
- 38.2k
3 votes
2 answers
337 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 ...
- 7,231
1 vote
0 answers
155 views
Tiling with a one-parameter family of non-congruent triangles
This post continues Tiling with triangles of same circumradius and inradius. The following are known about infinite sets of triangles that can be parametrized with one variable: from an infinite set ...
- 7,231
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 ...
3 votes
1 answer
123 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 ...
- 2,645
4 votes
1 answer
186 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. ...
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