For questions about properties and applications of triangles

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Integer Triangle Radicals conjecture

An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...
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108 views

Probability that one part of a randomly cut equilateral triangle covers the other without flipping

At Probability that one part of a randomly cut equilateral triangle covers the other, the case with flipping allowed was quickly solved. The case without flipping seems more difficult and hasn't been ...
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Similar Triangle dissections

Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$, $(d=1.15096...)$ that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The ...
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“Natural” labeling of triangles

The angles of a triangle are (capital) $A,B,C$ and the lengths of the sides are (lower-case) $a,b,c$. At your mother's knee, you were taught that the side whose length is called (lower-case) $a$ ...
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Area of $A'B'C'$ is to area of $ABC$ is $\frac{(m-n)^2}{m^2+mn+n^2}$

In the sides $BC,CA,AB$ are taken three points $A',B',C'$ such that $BA':A'C=CB':B'A=AC':C'B=m:n$.Prove that if $AA',BB',CC'$ are joined they will form by their intersections a triangle whose area is ...
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Proof of equilateral triangle given angles

Let's say we start with a scalene triangle ABC, with no given angle measures or side lengths: Then, we add 3 Isosceles triangles adjacent to this one, given that they have angle measures ...
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102 views

Number of triangles created after $n$ folds of a square

My daughter's grade 8 math homework included the following question. We were unable to find an answer, and I think we must have misinterpreted the question, as it seems way too hard. Fold a ...
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Howto prove that $\sum_{cyc}\cos\frac{A}{2}\cos\frac{B}{2}\le\frac{1+2\sqrt{2}}{2}+\frac{7-4\sqrt{2}}{R}r$

let $ABC$ is a triangle with inradius $r$ and circumradius $R$. Show that ...
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119 views

Number of distinct centroids of triangles formed by 40 equally spaced points on a the perimeter of a square

Suppose that we are given 40 points equally spaced around the perimeter of a square, so that four of them are located at the vertices and the remaining points divide each side into ten congruent ...
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25 views

How many different ways can a circle intersect a triangle N ways?

Consider a circle intersecting a triangle. The circle and triangle can have between 0-6 total intersection points. Is there a mathematical formula for the number of possible ways they can intersect ...
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Prove that the intersection point of lines $AK$ and $CL$ lies on the line $BO$

$AA', BB'$ and $CC'$ heights of an acute triangle $ABC$. The circle with center $B$ and radius $BB'$ intersects the line $A'C'$ in the points $K$ and $L$. Prove that the intersection point of lines ...
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Find the length of a triangle

Question is: Find the length of $\text{AO}$ and $\text{BO}$ My work, with the things I know already: Length: $\text{CO}=r$ and $\text{TO}=\frac{r}{4}$ and $\text{TG}=\frac{r}{2}$ and ...
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70 views

Minimize area of a triangle

Let $\Delta \mathrm\,{ABC}$ be a triangle in the plane and $X,\, Y,\,Z$ be points on sides $BC,\, CA,\,AB$, respectively. If lines $XY$ and $AB$ are not parallel, there is a location for $Z$ ...
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33 views

Catalog of triangles

Is there a catalogue of triangles in which one might find for instance the name of the right angle triangle with an angle of approx 35 degrees in which the altitude, median and side bisectors ...
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ABC is an acute angled scalene triangle.L,M,N are midpoints of the sidesBC,CA,AB.The perpendicular bisectors of AB and CA meet AL at D and E.

BD and CE cut each other at F inside the triangle.Prove that A,M,N,F are cyclic. I tried by taking a point on symmedian and also ...
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91 views

Maximum number of triangles given a number of lines.

This is just something I was wondering in my spare time. The problem: What is the maximum amount of triangles you can create given a fixed number of lines. The lines are of infinite length and can be ...
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543 views

Two circumcircles of triangles defined relative to a fixed acute triangle are tangent to each other (IMO 2015)

I'm posting here the question because I want to see a nice synthetic solution (not using complex numbers or inversive geometry) for the 3rd problem from IMO 2015. The problem is as follows: Let ...
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78 views

Alternative proof for the equality of two angles in an isosceles triangle.

From the answers of my previous question, I got an idea to prove equality of two angles in an isosceles triangle. In that question the equality of two angles in a right-angled-isosceles triangle was ...
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34 views

Generalization to higher dimensions of a statement about plane triangles

Let $\Delta=\Delta ABC$ be a plane triangle with area $F_\Delta$ and let $P$ be a point in $\Delta$. Draw lines through $P$ parallel to the sides of $\Delta$; then $\Delta$ is decomposed into three ...
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28 views

Statue and a flag distances

Next to a flagpole is a statue that measures 9m high. The upper end of the flagpole with the bottom of the statue form an angle of 53.13 degrees to the floor, and the upper end of the flagpole to the ...
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108 views

Calculate the area of a triangular field, knowing that two and 1 angle.

Hello so this problem came up while I was studying trig. and I seem a bit stuck: Calculate the area of a triangular field, knowing that two of its sides measure $80$ m and $130$ m and between them is ...
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58 views

How prove that $II^{\prime}< AA^{\prime}$ for $I $ and $I^{\prime}$ be their incenters?

Assume that we have two triangles $ABC$ and $A^{\prime}BC$. Let $I $ and $I^{\prime}$ be their incenters. How prove that $II^{\prime}< AA^{\prime}$? I have no idea how to do this, can this be ...
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95 views

Looking for an existing proof for a property of triangles

In my paper, I need the following lemma. I can prove it, but it is a little lengthy to be put inside the paper. I am wondering is there any existing proof that I can quote. Lemma 1: Let the nodes ...
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39 views

Lemoine Point triangle

from Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How can I ...
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Intesection point of feet of altitudes

If triangle has vertexes at $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, is the intersection points of feet of altitudes $$x_h = \frac{x_1x_2(y_2-y_1) + x_2x_3(y_3-y_2) + x_3x_1(y_1-y_3) + y_1^2(y_3-y_2) + ...
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229 views

How to easily prove Euler's theorem, $OI^2=R(R-2r)$?

If $R$ is the circumradius and $r$ is the inradius of some triangle $ABC$, with its circumcenter being $O$ and incenter being $I$, then how to prove: $$OI^2=R(R-2r)$$ I have seen many mentions of ...
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55 views

Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a ...
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106 views

Prove that the maximum volume of a triangular-base prism is $\sqrt{\dfrac{K^3}{54}}$ where K is the area of three triangles containing a vertex A

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and ...
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Proving there is no set of five distinct points s.t. every three points are the vertices of a right triangle.

We can see that the following proposition is true. Proposition : Each triangle $ABD, ACD, BCD$ is a right triangle for $$A(0,b,0), B(a,0,0), C(0,0,0)\ \ \ (a\gt 0, b\gt 0)$$ $\iff D$ is either ...
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101 views

Hijacked Malaysian plane position geometry

Sorry to get geeky in the midst of a tragedy and likely horrible crime, but does anyone know how they got this diagram showing the possible last known positions of the possibly hijacked Malaysian ...
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332 views

Triangle Packing-Problem

Theory and Question We define a normalized triangle $T$ as an ordered list of six points s.t. $p \in [0,1)$ for all $p \in T$. Let $T = [x_0, y_0, x_1, y_1, x_2, y_2]$ be a normalized triangle. We ...
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270 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
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217 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
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111 views

Minimize the perimiter of a triangle with an inscribed circle

A circle touches the two legs of an angle. How can one draw a line that intersects both legs, such that the circle lies within the triangle with as sides the two legs and the drawn line, and such that ...
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163 views

About the area of the region where the paper is twofold when you double a piece of paper in the shape of a triangle.

Suppose that you have a piece of paper in the shape of a triangle $ABC$ whose area is $S_0$ and that the area of the region where the paper is twofold when you double the paper in two along a line is ...
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135 views

Proving that the circumcenter is the centroid

Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
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113 views

triangles in a grid of $n\times n$ with positive coordinates

I need to count the number of triangles formed in a grid of $n\times n$ with positive integer coordinates $(0..n)$. For example for $n = 1$ the answer is 4.
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445 views

Euler's Line of a medial triangle

I have the following problem with a comment below on the steps that I took so far. Here is the example: Let triangle ABC be any triangle. The midpoints of the sides in Triangle ABC are labeled $A', ...
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An interesting geometry problem with angle bisectors and tangent

I have found the following problem: There is an acute $\triangle ABC$. Denote its circumcircle as $\omega$. The angle bisector of $\angle BAC$ intersects $BC$ and $\omega$ in points respectively $A_1$ ...
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How can I find the distance between two points within a triangle if I have the distance between each point and each vertex of the triangle?

Title says it all. It would be useful to extend the question to finding the distance if any of the points is outside of the triangle, but I'm trying to figure out the basic problem first.
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Is proof of Pythagoras Theorem using Similarity circular?

Please see this link: https://www.math.nmsu.edu/~breakingaway/Lessons/PTUST/PTUST.html (hope it dosent rot) Is this proof circular? I think similarity is proved by basic laws of trigonometry, ...
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Prediction Interval from Markov Chains

Thank you for taking the time to look at my question. Short, less involved question: How do you find Prediction Intervals with non-Gaussian residuals? Here is the situation: I have made a model that ...
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Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem ...
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Inequalities of the triangle

I created a inequality below, what you think guys? Let $ \alpha',\beta',\gamma'$ be angles of an acute triangle, and let $ n\in\mathbb{N}^{*}$ and $\displaystyle j\in\mathbb{N}$. Prove that: $$ ...
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Angles sum in a triangle on the x- axis.

$\angle BCA=90$ degrees I probably do not understand the concept of angle sum in a triangle but here is the thing. $\angle BAC$ is negative by convention. So is $BCA$ going to be greater than 180 ...
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Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
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Joint density of Triangular RV and Maximum of Triangular RVs, parameterised by Uniform RV

Let $x$ be drawn from the uniform distribution on $[0,1]$. $x$ parameterises the Triangular distribution $Y$ with support $[0,1]$. I.e., $$ f_Y(y_i | X = x) = \begin{cases} \frac{2y_i}{x} \quad ...
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The angles of the triangle given the position vectors of the triangle using the scalar product

I can't seem to show for question (a), I am not sure if it's because of my wrong calculation, or is it the question has the wrong values? What I have done is I used the scalar product. I found that ...
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29 views

From a point to the Vertex

I was aked to solve the following problem: Guiven three lenghts and a triangle ABC, from every vertex whe draw one of the three lenghts, find the conditions such that the three lenghts meet in one ...
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Determine third point of Right Triangle when two points and all sides are known and $A\hat BC=90$

I have two points and all sides of right triangle I need find A point \begin{gather*} |AB| = 1 \\ |BC| = 1 \\ |AC| = \sqrt{1^2 + 1^2} = \sqrt2 \\ A(?,?) \\ B(0,0) \\ C(1,0) \\ \\ |AB| = 1 \\ |BC| = ...