For questions about properties and applications of triangles

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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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73 views

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| = ...
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23 views

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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301 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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46 views

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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74 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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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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174 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 A(...
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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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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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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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346 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 the ...
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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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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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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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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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127 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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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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153 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 didn'...
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284 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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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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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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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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593 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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Discovering length of line

I'm attempting to work out length of BD from below diagram : The length of BD is -2 +- some value. But since I do not know the y co-ordinate of B can the length of BD be determined from ...
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An geometry problem. proved that the circles inscribed in triangle ABD&CAD are tough each other.

The inner circle of triangle $ABC$ touches $BC$ at $D$ . Show that the circles inscribed in triangles $ABD$ and $CAD$ touch each other.
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right-angled triangle problem

If the hypotenuse is 8 cm, one of the sides is X cm and the other 4 cm longer how do i find the two unknown sides? I started by applying the Pythagorean theorem like this $x^2+(4x)^2=8^2$ but i don't ...
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Calculate the percentage of a triangle inside a cuboid?

I have a large (order 10^7) collection of triangles in 3D space. I also have a cuboidal mesh also of order 10^7. For each triangle I need to calculate the area of that triangle which is inside any of ...
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Angles and How they Correspond to Sides of a Triangle

I have done some Googling but am not sure what question I need to ask. I came here instead. I am at a 9th grade math level (Geometry) and working on a problem that is asking me to find x and y based ...
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Why we name one side as the perpendicular of an angle but does not actually define it?

If I have a right angled triange: $\qquad \qquad \qquad \qquad$ I was wondering why we name the sides like this? The base of $A$ kind of make sense. But the perpendicular of $A$ what relation does it ...
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Construction of a triangle using symmetry

I need to construct a triangle $\Delta \textrm{ABC}$ knowing that $t_a = AS$, $|AS| = 6\, cm$, $|\measuredangle \textrm{BCA}| = 30°$ and $|AB| = 5.5 \,cm$. I've been told that it's possible to do it ...
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32 views

Area of a triangle on an Argand diagram

I am working on two problems: 1) Find three distinct roots of the equation $8z^3 + 27 = 0$ I solved this and ended up with \begin{align*}z_1 &= \frac32 \left( \cos (\pi/3) + i\sin(\pi/3)\...
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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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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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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 ...