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3
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50 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 ...
3
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71 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 ...
3
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256 views

How to calculate Fermat point in a triangle most efficiently?

I am aware of this question, but mine is a bit more specific. I want to find the coordinates of the Fermat point for a given triangle. Assuming that no angle in the triangle is larger than 120 ...
2
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41 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 ...
2
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105 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 ...
2
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87 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 ...
2
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0answers
162 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 ...
2
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56 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 ...
2
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66 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 ...
2
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153 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 ...
2
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110 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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352 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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0answers
45 views

Rationality in Triangle

How can I justify this answer? I think the answer is infinite, but cannot justify it///
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29 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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17 views

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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0answers
27 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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0answers
24 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 ...
1
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56 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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36 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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0answers
41 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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12 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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58 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 ...
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0answers
142 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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0answers
73 views

Question on Proof of Shoelace Formula

I was looking for a way to prove the shoelace formula when I found this proof: For this clockwise order to make sense, you need a point O inside the polygon so that the angles form $OA_{i}A_{i+1}$ ...
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0answers
35 views

Largest possible value of a side

ABC is a triangle with side a, b,c with $a\geq b\geq c$ and $sin^3A+sin^3 B+ sin^3 C=a^3+b^3 +c^3$ How do I find the largest possible value of a? I tried to use the law of sines ratio, but it ...
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123 views

Finding Areas in triangles using ratios

What theorem/theorems should be used to find the shaded area? Y and M lie on the sides Ab and Bc respectively of the triangle YMB such that AY/MI= 1/4 and O/M = 1/3. Area ABC=35 PC and QA intersect ...
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0answers
104 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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0answers
82 views

maximum length of a scaled vector in a triangle (simplex)

Given a triangle (or, in general, a simplex) $T$ and a vector $\vec{s}$, I'd like to compute the quantity $$ \max\{|x-y|: x,y\in T, x-y = \alpha \vec{s}, \alpha\in\mathbb{R}\} $$ i.e., the maximum ...
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0answers
35 views

Two coloured plane

Can you prove that For any two angles $θ,ϕ$ there exists a monochromatic triangle that has angles $θ,ϕ,180−(θ+ϕ)$ in two coloured plane?
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65 views

How to find the inverse position inside a triangle

If i were standing in a triangle - How do i calculate the inverse of my position? Can it be done? It's easy inside a rectangle, but I can't think of how you would do it inside of a triangle. I'm ...
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41 views

Triangular exponentation logarithm and inverse

The generalized formula of triangular exponentation on real numbers field is $x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)} $ It's my ...
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0answers
440 views

General formula for computing triangular gaussian quadrature.

While this is a simple question, I'm totally lost. Is there any general formula for generation of n-point gaussian quadrature over a triangle? I'll use this formula to generate a variable-point (7, ...
0
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29 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
25 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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28 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
17 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 ...
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0answers
13 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
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0answers
52 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
17 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
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0answers
49 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
36 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.
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36 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
36 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 ...
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0answers
18 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 ...
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0answers
24 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
77 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
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0answers
31 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
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0answers
34 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
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0answers
23 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
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0answers
27 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 ...