For questions about properties and applications of triangles

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Right triangle and Sine function?

Given two angles and the hypotenuse of a right triangle, when trying to find the length of the side opposite the given angle, why and how does it's angle and supplementary angle yield the same answer? ...
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283 views

Line tangent to circle inside an isosceles triangle

If you take a circle enclosed inside an isosceles triangle, and then draw a line which is tangent to the circle and which intersects with the two equal sides, is that line parallel to the triangle's ...
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Proof of Menelaus using areas

I've tried to proof Menelaus' theorem using areas, but I've didn't figure out how. Some suggestions would be appreciated. Menelaus' Theorem states : Given a triangle ABC and a transversal ...
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47 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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71 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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Finding maximum subset-triangles

To a given base (ab), triangles are constructed by choosing a point (p). How can i find the maximum subset-triangles(*)? (*)subset-triangle: p' is inside the triangle abp. Allowed interceptions from ...
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43 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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170 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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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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95 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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29 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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48 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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325 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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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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50 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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$PC+PD$ is least when the angles $CPA$ and $DPB$ are equal

$C$ and $D$ are two points in the $same$ side if a straight line $AB$ and $P$ is any point in $AB$. Show that $PC+PD$ is least when the angles $CPA$ and $DPB$ are equal No idea how to solve this ...
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75 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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77 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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69 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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30 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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124 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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48 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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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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144 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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44 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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277 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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90 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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39 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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46 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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586 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, ...
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27 views

How to determine if a triangle is inside another triangle without any intersecting sides

This question is for getting the right logic down for a programming task. I need to be able to determine if a triangle is located inside another without any sides intersecting each other. The two ...
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If the circumcircle of a triangle cuts its nine point circle orthogonally,then prove that $\cos A\cos B\cos C=\frac{-1}{2}$

If the circumcircle of a triangle cuts its nine point circle orthogonally,then prove that $\cos A\cos B\cos C=\frac{-1}{2}$ I know that two intersecting circles are orthogonal if any one of the ...
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14 views

Help in finding angles in free body diagram.

I figured that $$\tan^{-1}16/8=36.87^\circ$$ but can't find which property of triangles was used to find $53.13$ degrees and $73.74$ degrees angles.
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Relative velocity- Finding the direction of wind.

An aircraft is flying due south at $350~\text{kmh}^{-1}$. The wind is blowing at $70~\text{kmh}^{-1}$ from the direction of $\theta$, where $\theta$ is acute. Given that the pilot is steering the ...
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How to find the third point of a triangle in a 3D space (arm rig)

I am attempting to create a system that will replicate arm movement, so far I have mastered this in a 2D plane however I am having trouble adding the third dimension. Here is what is given, You know ...
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12 views

A new family circle associated with the Tucker hexagon and the Symmedian point

I am looking for the problem following: Let ABC be a triangle, let $A_1B_1C_1$ be a cevian triangle of the symmedian point. Let $B_aC_aC_bA_bA_cB_c$ be a Tucler hexagon of $ABC$. Such that $A_bA_c ...
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A generalization of the first Droz-Frany circle

I am looking for a proof of the following problem: Let $ABC$ be a triangle with circumcenter $O$, and the medial triangle $M_aM_bM_c$. Let $O_a, O_b, O_c$ be three points on three lines $OA, OB, ...
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Sampling from Irwin-Hall distribution using triangular distribution

So I need to sample from the Irwin-Hall distribution using rejection sampling with the triangular distribution. I built 2 functions: The first is d_irwin which receives an $x\in supp(g)$ and the n we ...
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64 views

Calculate the coordinates of two points in an isosceles triangle

In the triangle below, given the point A, angle θ and length d of the two equal sides, how can the points B and C be calculated? Edit:After brainstorming for quite some time, I ended up with a ...
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Area of non-spherical triangle on a sphere

This is a followup to the question Area of triangle on a sphere (not spherical triangle) Since it's now almost two years later, I'm making it a new question. The problem is to find the area of ...
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Division of Solid Angle When Subdividing Spherical Triangle

Suppose I have a spherical triangle (no special properties; in particular, not equilateral) with a known solid angle. Now, I divide it into four new spherical triangles by bisecting each edge: ...
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29 views

Congruence of triangles: SSA criteria

It is well known that this criteria does not work in general. I am trying to answer to the following question if two triangles have two sides and the angle NOT between them equal, they are either ...
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20 views

Prove that the area of a triangle DEF is correct.

There's any triangle ABC. First player 1 has to set D on AB so that in the end the triangle DEF has the highest possible area. Second player 2 has to set E on BC so that in the end the triangle has ...
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44 views

How to find a triangle's perimeter only using base and height?

Without measuring the length of the other two sides, is there a way to find the perimeter with one side (Base) and the height of that side?
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Prove that angles are equal using complex numbers

that in triangle $\Delta ABC$, where $D$ is a point on side $BC$, $E$ is a point on side $AB$, $BD=AC$, $AD=AE$ and $AB^2=AC\cdot BC$: angle $BAD$ is equal to angle $CEA$. This problem can be quite ...
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Producing a 3D Net from a 3d inspired image

Producing a 3D Net from a 2D Image I'm trying to find the volume of the illustration, I've taken reference from the medium size of a strawberry's diameter, I've applied this scale to the remaining ...
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110 views

Proof for Pappus's Centroid Theorem with basic geometry?

How to prove Pappus's Centroid theorem about volume for a triangle rotated around an external axe? The theorem says that the volume V of a solid of revolution generated by rotating a plane figure F ...
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31 views

Using similar triangles, find l?

I am attempting to render some 3D shapes and am having trouble with some of the math. I'm hoping someone can help me out here. Given two arbitrary points in space, P and C, I need to find l. I have L ...
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73 views

Proving triangle inequality using complete-linkage between clusters and arbitrary dissimilarity measure

Assuming a dissimilarity measure d satisfies the usual properties, I need to prove that complete linkage ( i.e. d(A,B)=maxx∈A,y∈B{d(x,y)} ) either satisfies or does not satisfy the triangle inequality ...
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Find the Radical Axis of the Circumcircle of Triangle ABC and its Nine Point Circle

Given a triangle ABC, find the radical axis of its circumcircle and its nine point circle.