For questions about properties and applications of triangles

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7
votes
1answer
134 views

Can the $9$ point circle be generalized to $n$-gons of $n\gt3$?

All triangles have concyclic vertices and have a $9$ point circle which intersects the triangle's feet and the midpoints of its sides (as well as $3$ other significant points). Is this special for ...
5
votes
1answer
66 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
5
votes
1answer
89 views

explaining the resriction $b<a<2b$ in a triangle

I saw in a book that if $ABC$ is an isosceles triangle $(AB=AC)$ and the triangle is tangent to a circle in points $D,C$ and $AC$ is intersecting the circle in point $E$; $AC=a$, $BC=b$ so it has ...
3
votes
1answer
69 views

Ortocenter and incenter

In triangle $ABC$: $H_{1}$ is a foot of an altitude from side $BC$, $H_{2}$ is a foot of an altitude from side $AC$, $H_{3}$ is a foot of an altitude from side $AB$, $M_{1}$ is midpoint of $BC$, ...
3
votes
1answer
325 views

Geometry - optimal 2D mesh between X expendable points

Say you have X points on a plane. If you connect two points, you form a line. Connecting three points forms a triangle. A line cannot cross a line, and a smaller triangle cannot be created inside a ...
2
votes
1answer
68 views

How $\frac{\cos \alpha_1}{\sin \alpha}+\frac{\cos \beta_1}{\sin \beta}+\frac{\cos \gamma_1}{\sin \gamma}\leq\cot \alpha+\cot \beta+\cot \gamma$

Let are any two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$. How prove that $$\frac{\cos \alpha_1}{\sin \alpha} + \frac{\cos \beta_1}{\sin \beta}+ \frac{\cos ...
2
votes
1answer
31 views

How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - \cot A)$?

Consider all triangles $ABC$ where $A < B < C \leq \frac{\pi}{2}$. How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - ...
2
votes
1answer
74 views

Trigonometric Substitution and the Triangle Inequality

I was reading the solution to this problem: If $x, y, z$ are real numbers and $x+y+z=xyz$ then $x(1 − y^2 )(1 − z^2 ) + y(1 − z^2 )(1 − x^2 ) + z(1 − x^2 )(1 − y^2 ) = 4xyz$ The solution is to ...
2
votes
1answer
23 views

tetrahedron height

I've got the next figure: Now I would like to calculate the height, so from D to the plane ABC. First, I've tried with a coordinate system, but it's to difficult to take these distances into ...
2
votes
1answer
75 views

Find direction, angle or co-ord of unknown vertices using only distance?

My current issue is that I have a triangle, where I know all the line distances as well as an origin coordinate. Is there any way I can then gain the coordinates of the other vertices with this ...
1
vote
1answer
37 views

Euclid I.24 Proof Why is DFG greater than EGF?

Proposition 24 If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the ...
1
vote
1answer
22 views

How Many Triangles are Created by n Lines in the Plane?

Suppose we are given n lines in the plane in "general position", which in the present case we define to mean the following: A. no 2 lines are parallel or identical B. no 3 lines have common ...
0
votes
0answers
24 views

Triangle Section Side Lengths

Point $D$ is on side $BC$ of $\triangle ABC$, with $AB=3$, $AC=6$, and $\angle CAD = \angle DAB = 60 ^{\circ}$. What is the length of $AD$?
0
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0answers
22 views

Equilateral Triangels - geometry- minimum sums

In the following figure, the triangle ABC is arbitrary and so is the point P in its interior. We construct the two equilateral triangles APE and ABD. Show that PA+PB+PC=DE+EP+PC. Conclude from here ...
0
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0answers
18 views

Determine Angle based on Vertical Displacement

I need a formula that will help me identify a observing angle based on the following example: Launching a bottle rocket. Test 1: looking from 35 degree angle, when a bottle is launched from ground, ...
0
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0answers
42 views

Geometry Problem relating similarity.

Given a triangle $ABC$ and $D$ be a point on side $AC$ such that $AB=DC$, angle $BAC=60-2x$, angle $DBC=5x$ and angle $BCA=3x$ prove that $x=10$. Source: 150 Nice Geometry Problems - Amir ...
0
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0answers
25 views

Create dynamic cities of perspective angle x

I'm creating a tilemap... I found you can create unique building sizes with perspective with six tiles using parallel projection, whose angles are always 45 degrees... this allows you to connect to ...
0
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0answers
44 views

Trigonometry, find distance of arc movement

Imagine I have the setup as follows: I want to compute the height x in State 2, depending on how much the blue axis have moved. That is, the distance ...
0
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0answers
36 views

Find angle of an arc in the circle using 3 coordinates

I want to find angle of semicircle. I have 3 coordinates (center_a,center_b) , (pivot_a,pivot_b) and (last_point_a, last_point_b). From triangle , i can find angle using equation using cosine ...
0
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0answers
44 views

Similar triangle, Quick question (Thick Lens Formula)

http://www.panohelp.com/thinlensformula.html On the right hand side, f is defined as focus of the lens, i understand why the image distance is (f + fm). However i have spent an afternoon and could ...
0
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0answers
19 views

Rationalizing triangle relationship for a bar inside a hemisphere

Find ratio of length AE to diameter of the hemisphere Given that ABD = 90 degree, AO = BO, O is the center of the sphere. This is actually a physics problem, but I bump into geometric problem ...
0
votes
0answers
22 views

Rotating a triangle in different coordinate systems.

My android application uses openGL. OpenGL coordinate system has the origin in the middle and goes from -1 to 1. When I am rotating an equilateral triangle in the openGL coordinates, the triangle ...
0
votes
0answers
72 views

Calculate height from two right angled triangles sharing an edge

I am trying to calculate the perpendicular distance of a unicycle-like robot from a wall using two successive measurements from an ultrasonic sensor. The problem is modelled as shown: (EDIT). The ...
0
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0answers
25 views

Solving ray/triangle intersection - comparison on methods

In 3D I have a ray $q + t\vec{d}$, and a triangle with vertices $a$,$b$,$c$. I want to find the parameter $t$. I have solved this by noting that a vector in the triangle plane is perpendicular to the ...
0
votes
0answers
149 views

Finding general Cartesian coordinates of the third vertex of a triangle lies between two circles

I'd like to find the Cartesian coordinates of the vertices $(a, b, c)$ of the triangle $T$ inscribed in the circle $S^1$ and circumscribed about the circle $D$ ? I start my calculations as follows: ...
0
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0answers
48 views

minimum sum of distances from vertices

Find a point on the plane of a triangle such that the sum of its distances from three vertices is minimum. I am guessing that it is the centroid but I can't prove that.
0
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0answers
44 views

Moving up the Y axis the lengh of the hypotenuse of a right triangle

If i have a right triangle ABC with B being the right triangle and length AB = 50 and length BC = 50. Based on the Cartesian coordinate system if i wanted to move up the Y axis the length of the ...
0
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0answers
37 views

Complex Number and Geometry

Given $A(3+4i)$, $B(-4+3i)$ and $C(4+3i)$ be the vertices of a triangle $ABC$ which is inscribed in a circle $S=0$. Let $AD, BE, CF$ be altitudes through $A, B, C$ which meet the circle S=0 at ...
0
votes
0answers
29 views

Volume of a Part of a Triangular Prism Enclosed in a Sphere

I'm having trouble finding the volume of the shaded prism. I know how to calculate the volume by extending the height of this prism to create a triangular based pyramid, but I cannot get the same ...
0
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0answers
38 views

Proof metric space with distance function

Thats the first time i have to do such an proof but don't know how, never seen or done this before. Especially (iii). Let $X$ be the Set of all complex sequences. $$ d((a_n),(b_n)) := ...
0
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0answers
122 views

Generate X, Y, Z coordinates of 3D triangular prism with Edge Rounding

I'm trying to create an interactive 3D visualization with Python and Mayavi for inputs to an analysis program. The program accepts certain primitive shapes which it combines (constructive solid ...
0
votes
0answers
37 views

Solving for and x,y,z coordinate in a 3D plane

This is hard for me to explain, but basically I am making a game and I want a 3rd person like camera. I have a lot of information about how the camera should be but I can't seem to get the camera to ...
0
votes
0answers
46 views

Optimally connecting 2D points to form as many nested triangles that do not overlap

So I have 3 cities that are pretty far apart (2D plane). Distance between each of them more than 50km. In every city I have nodes scattered throughout the city in a random fashion. I know the 2D ...
0
votes
0answers
39 views

Force to change the base length of an isosceles triangle

Given an isosceles triangle with legs 7' long weighing 160lbs. What horizontal force would be required to change the base width from 15' to 13'? The ends are on wheels-so assume perfect conditions ...
0
votes
0answers
28 views

How to prove that PH is containing midpoint of side MN from this circle and triangle problem?

Given: triangle ABC is acute triangle. M and N are midpoints of AB and BC respectively, while BH is altitude of triangle ABC. Circles AHN and CHM meet at point P. (P is not same with H) How to ...
0
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0answers
24 views

How to prove that P,G, and K are collinear from this triangle problem?

Given: triangle ABC. We choose point Q at AC, P1 and P at BC, and R at AB, such that: AR/RB= BP/PC= CQ/QA= CP1/P1B Suppose G is centroid of triangle ABC, and K= AP1 ∩ RQ. How to prove that P,G, and ...
0
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0answers
34 views

I stumbled when i saw this (Travelling Salesman related)

I have here 5 locations like below I then have a tour of 1,2,3,4 (point 5 isn't inserted into tour yet) like below I then find the shortest addition distance to include point 5 into tour, the ...
0
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0answers
15 views

How to prove that angle EDL is same with angle ELD in the following triangle problem?

Given: triangle ABC with angle A equal with 60 degree. We choose points D and M at AC, points E and N at AB, such that DN perpendicular to AC and EM perpendicular to AB. If L is midpoint of MN, ...
0
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0answers
47 views

Triangle rotating freely around origo, need to calculate corners.

Lets say I have a triangle with corners $(-1, -2)$, $(0, 2)$ and $(1, -2)$. I specify a line that is exactly one side of the triangle, for example $(-1, -2)$, $(0, 2)$. Now, I rotate the triangle ...
0
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0answers
47 views

Need explanation for clustering coefficient formula

I need some explanation for clustering coefficient formula itsef firstly and why it can be used for detecting communities in a social network! Also I would like to know why it is not a good method for ...
0
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0answers
63 views

Ratios in a rhombus

NOTE: I am NOT looking for a full answer,just a hint. Last problem on this question. BdMO 2013 Chittagong: Let $ABCD$ be a rhombus.Let $G$ be a point outside the rhombus such that GE is ...
0
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0answers
50 views

Triangle inscribed insemicircle area-ratio question

My approach: $m<A= 60$ degrees and $m<C=30$. This creats a 30, 60, 90 triangle with ratios $$1:\sqrt3:2$$ After getting the ratio's of the areas, I obtain $$\frac{b*h}{\pi r^2}=\frac{1*\sqrt ...
0
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0answers
72 views

Fastest way to check whether the triangle inequality is satisfied

If we are given the lengths of the three sides of a triangle, and we simply add the 2 smallest sides and check to see if the sum is larger than the third side, will this always yield the correct ...
0
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0answers
113 views

Line Triangle Intersection Mathematics

I am following the math in the book Real Time Collision Detection by Christer Ericson. On pages 184 thru 188, he discusses how to test for an intersecting line against a triangle. I replicated the ...
0
votes
0answers
104 views

Sum of angles in a hyperbolic triangle with one ideal angle

I want to calculate the sum of the angles of the triangle formed in the hyperbolic plane from the points $(-1,1), (0,1)$, and $(1,1)$. This forms an angle at the origin which has an infinite slope for ...
0
votes
0answers
149 views

Determining a point in 3D space

So given a point, a rotation around the y-axis, a rotation around the x-axis, and a distance, how can one calculate the relative point in space? For example, the beginning coordinates are (0,0,0). ...
0
votes
0answers
231 views

pixels in a projection of a triangle in 3d space onto a 3d plane through a pinhole camera

I have a triangle in 3d space. The x and y components of its vertices make a 2d right isoceles triangle. I am projecting it through a pinhole onto a plane. The projected triangles on the plane are now ...
0
votes
0answers
160 views

Get value of angle with 45 degrees as maximum and 0 and 90 degrees as minimum

I want the calculate the "value" of an angle in such a way that: The angle of 45 degrees corresponds with the maximum value of 1 The angles of 0 and 90 degrees correspond with the minimum value of 0 ...
0
votes
0answers
207 views

Uniform Random Points on a triangle using only edge plane normals

For a triangle $ABC$ in 3D (each point has x, y, z coordinates) is it possible to generate uniform random points on the triangle from only the following data: Normal of the triangle plane $N = ...
0
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0answers
230 views

Problem finding a $2$D point in a triangle

I have a triangle with $3$ Points - $A$, $B$ and $C$ and the angle $\alpha$. $A$ and $B$ are fixed. $C$ is any point at the side of '$b$', and $\alpha$ has at $A$ and $B$ the same size. I need to ...