Questions tagged [triangles]
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129 questions
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The Snellius-Pothenot problem and antigonal conjugates
The Wikipedia article on the Snellius-Pothenot problem has a section titled "Geometric Solution" that contains a major mistake. It says "By the inscribed angle theorem the locus of ...
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"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 ...
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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, ...
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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 ...
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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 ...
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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?
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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.
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
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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:
$$
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Japanese-style geometry problem: seven circles in a triangle, show that one is congruent with another two
I made up the following Sangaku problem.
In the diagram below, circles of the same color are congruent. Wherever things look tangent, they are tangent.
Show that the red circle is congruent with the ...
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Is there a simpler proof for this simple geometrical result? (An equilateral triangle contains three congruent circles, prove two lengths are equal.)
In equilateral $\triangle ABC$, $D$ is on $AB$, $E$ is on $AC$, and the incircles of $\triangle ADE$, $\triangle DBE$ and $\triangle EBC$ are congruent.
Prove that $BD=DE$.
I asked this question on ...
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Conjecture on a cyclic quadrilateral associated with central line of triangle
Using computer I found a conjecture as follows (click to check by geogebra):
Conjecture: Let $A$, $B$, $C$, $D$ lie on a circle such that $A$, $B$, $C$, $D$ are not formed an Isosceles trapezoid. ...
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On 3 centers of triangles
We add a little to On two centers of convex regions
For any interior point $P$ in a planar convex region $C$, we define d_max to be the maximum distance from $P$ to the boundary of $C$ (distance from $...
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Dissection proof of Heron's formula?
In his recent book, Love Triangle, Matt Parker playfully complains that Heron's formula is an "opaque formula, and I feel like you just chuck in the side-lengths, turn a series of arbitrary ...
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Question on a vector inequality
Is it true that
$$
\min\left( \begin{aligned}
&\|\mathbf{u}\| + \|\mathbf{v}\| - \|\mathbf{u} + \mathbf{v}\|, \\
&\|\mathbf{u}\| + \|\mathbf{w}\| - \|\mathbf{u} + \mathbf{w}\|, \\
&\|\...
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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 ...
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Question on a min inequality
Is it true that
$$
\min\left(a^2 + b^2 - \sqrt{a^4 + b^4 + 2a^2b^2\cos(x)}, b^2 + c^2 - \sqrt{b^4 + c^4 + 2b^2c^2\cos(x-y)}, a^2 + c^2 - \sqrt{a^4 + c^4 + 2a^2c^2\cos(y)}\right) \leq \frac{1}{3}
$$
...
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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 $G$.
...
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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.
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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?
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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/...
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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 ...
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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 ...
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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 ...
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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., ...
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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 = ...
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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 ...
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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 $x+2y\neq 0\neq y+2x$. ...
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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-...
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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 ...
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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 ...
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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 ...
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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)....
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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 ...
- 19.6k
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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