For questions about properties and applications of triangles

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Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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35 views

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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41 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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65 views

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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160 views

History of incenter and Euler line

It is easy to see that if a triangle is isosceles, then its incenter lies on its Euler line. Who first proved the converse of this result and what technique was used? (See the post "The incenter and ...
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115 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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465 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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47 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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32 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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20 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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41 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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82 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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33 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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45 views

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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54 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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81 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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112 views

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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99 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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261 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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214 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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142 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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85 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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161 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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112 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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414 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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36 views

What can be said about triangle with certain condition?

This question comes from 1988 Irish Mathematical Olympiad, for all those interested. A mathematical moron is given the values $b,c,\alpha$ for a triangle $ABC$ and is required to find $a$. He does ...
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24 views

Given a single point in 3d space, and 3 points that make up a triangle, find the closest point in/on the triangle to the point.

Given point $(p,q,r)$ and 3 points which make up a triangle, find the closest point in the triangle to the point in space. From the triangle, we can find the equation of the plane $Ax+By+Cz+d=0.$ ...
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71 views

Abc is a triangle

Abc is a triangle (drawing of the triangle with measurements up the side of each side) Make a full size drawing of triangle abc in the space below The line AB has been drawn for you. Leave in all ...
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18 views

Calculating pairwise distance of two N-dimensional vectors given their length and angle

I am not a mathematician, so apologies in advance for any nomenclature blasphemy. Given the magnitudes of two vectors $b$ and $c$ and the angle between them $A$, I can calculate their distance in 2-D ...
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127 views

Cabri 3D - Rotating a triangle

I'm given the exercise, in Cabri 3D, to rotate the triangle T around the axis AB and lead it via the triangle To to the triangle T'. I tried to rotate the triangle T around a fixed point and then ...
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52 views

Concave triangle?

I know that in Euclidean and (I think) Spherical geometries don't have concave triangles, but is there any set of axioms that would allow this?
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35 views

Finding the coordinates of the third point in a triangle using simultaneous equations

I have 3 circles A,B & C that touch each other at tangents. The centre points of these 3 circles are to be joined to create a triangle. I know the coordinates of 2 of the circles centre points ...
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40 views

Intersections of convex hulls

Given a set of $n$ points $\{A_1, \ldots , A_n\}$ of the plane and every possible triangle formed with $3$ points $A$, I would like to describe the intersections fo theses triangles. By intersection, ...
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23 views

Finding other two vertices when one vertex and each point on the triangle is known ?

I am working on some gesture recognition for my game. I am stuck on a problem. I have one vertex i.e the starting point and every point on the triangle, I also have the centroid. So how do I find the ...
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43 views

What's wrong with my reasoning while setting up a limit?

I was writing an answer to this question, which asks about what happens to the apex of an isosceles triangle if a vertex is at infinity. I thought it would be very easy to prove it by setting up a ...
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144 views

How to find mass points and ratios in a triangle?

How to find mass points with weights and ratios is my question. In my class, we learned about mass points. First we had the given ratios of 2 side lengths. Given: MC = d MB = e MA = f BD:DA = ...
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26 views

Triangles with vertices on conics and their foci

Let $A$, $B$, and $C$ be the lengths of the three sides of a triangle. Let $α$, $β$, and $γ$ be the measures of the angles opposite those three sides respectively. Mollweide's formula tells us that ...
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46 views

Issue with a right-angled triangle

The area of the right angle triangle is $18\text{ cm}^2$ and the ratio of its legs is $2:3$. What is the length of the hypotenuse? I assumed the lengths of two sides to be $2x$ and $3x$. I used ...
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141 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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22 views

Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
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55 views

Closest Points on Two Triangles in 3D Space

I have two triangles in 3D space, defined by 3 (x, y, z) points each. I'm looking to find the closest points between the two triangles, whether that be on surface, edge, or point. I'm unsure how to ...
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How to find the length of the union of Isosceles triangles

I am given N number of right angles triangles all of which are also Isosceles triangles. For each triangle, I am told where they start on a number line and where they end on a number line with end ...
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41 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
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29 views

How to prove that $FC/FA + GC/GA= 0$ from this triangle problem?

In triangle $ABC$, a transversal line intersects $AB$, $BC$, $CA$ at $D,E,F$ respectively. $BS$ intersects $AC$ at $G$, where $S$ is the intersection of $AE$ and $CD$. How to prove that ...
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98 views

How to prove these equations base on this following interior and exterior angle bisectors problem?

In the triangle $\triangle ABC$, length of $BC$ is larger than length of $AC$. The interior angle bisector of $\angle C$ intersects $AB$ at $D$; and the exterior angle bisector of $\angle C$ ...
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93 views

solve this complex triangle question ?

,D,E,F are midpoint of triangle ABC on sides BC , CA , AB. The feet of the altitudes from A,B,and C are P,Q and R. h is the orthocentre and O is the circumcentre . Then prove 2OD=AH. The nine ...
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44 views

Ratios of right triangle integer multiples to PI

It is known that in a right triangle with angles 30 and 60 degrees the cathetus at the 60 angle is equal to the 0.5 of hypotenuse. In other words an angle with cosine 0.5 is equal to PI/3. Is there ...
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14 views

3-D evaluations of a triangle

We all do evaluations of triangles on 2-D space based on the fact that the sum of its internal angles is 180 degree. When we draw a triangle on a sphere this sum changes and gets bigger than 180 ...
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95 views

Area of a triangle using vectors

I have to find the area of a triangle whose vertices have coordinates O$(0,0,0)$, A$(1,-5,-7)$ and B$(10,10,5)$ I thought that perhaps I should use the dot product to find the angle between the ...