For questions about properties and applications of triangles

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1answer
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Problem finding coordinates in a earth like coordination system

A picture with the problem Hey guys Given: two coordinates $A(a_1,a_2), M(m_1,m_2)$ , the distance between $B$ & $C$ is known as $w, d(B,C) = w$ d(B,M) = d(M,C) where d is the great-circle ...
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1answer
19 views

Determine vertex coordinates of a triangle if length and angles of opposite are known

Given a triangle such as this: Where $C$, $A$ and $B$ are cartesian coordinates and $a$, $b$, $c$ are the lengths of the sides. I know that $$C = (b\cos\theta,\;b\sin\theta)$$ where $\theta$ is the ...
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0answers
38 views

Question to prove triangles Equilateral on provided conditions

Let D, E, F be points on the sides $BC,\ CA,\ AB\ $respectively of a triangle $ABC$ such that $BD = CE = AF\ and \ ∠BDF = ∠CED = ∠AFE$. Prove that $ABC$ is equilateral.
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3answers
61 views

Placing Pandas in a Triangle Pen

I am working on a bit of a silly problem in my introductory discrete mathematics course. I have five pandas that I need to place in a pen, and I have a pen that is the shape of an equilateral triangle ...
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3answers
26 views

How would I solve for the bases of an isosceles without the height?

In the image below if you were to take 36 degrees and set it between 0-180 degrees how would you solve for the bases between points AC. There are special cases for the bases length such as (60 degrees ...
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1answer
42 views

Area of Triangle in ellipse

Full question: Prove that the area of the triangle formed by three points of an ellipse, whose eccentric angles are $\theta , \phi$ and $\psi$ , is ...
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2answers
46 views

Show that the perimeter of an octagon is $8(\sqrt{2} -1)$

My question is as follows: The top of a table is made in the shape of a regular octagon by cutting the congruent isosceles triangles from the corners of a $1$ m square piece of wood. Show that the ...
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4answers
71 views

How can an angle be negative?

How can be angle be negative like sine(-60) , cosine(-50) ? Which quadrant do they fall if we have the negative angles ? I dont see any negative angles in full ...
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1answer
44 views

Common meeting point for 3 points to reach 4th point [closed]

Problem statement: We are 3 friends at 3 different locations $A, B, C$ and want to reach a location $D$. Each person will take a separate cab to a common meeting point $E$, and then take a single cab ...
2
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3answers
43 views

What does a triangle become?

I'll get to the point. Imagine a right triangle in 2D. If you move one of the points (not the 90 degree vertex) very far away, then the corresponding angle will become smaller as the point moves ...
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1answer
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In a triangle $ABC$,$AD,BE,CF$ are the altitudes and $R$ is the circumradius,then find the radius of the circle $DEF.$

In a triangle $ABC$,$AD,BE,CF$ are the altitudes and $R$ is the circumradius,then find the radius of the circle $DEF.$ This triangle is not given to be equilateral or anything else.Only the three ...
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1answer
26 views

Help With Steps Of Finding Orthocenter

I'm trying to find the orthocenter of $M(-8,0)$, $N(0,0)$, $P(-4,6)$. I thought I did all of the steps right but I keep getting an answer of $(-4,6)$, but my book says $(-4,2.6667)$. Here are the ...
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1answer
40 views

Line $x=-1$ is side BC of equilateral triangle ABC circumscribing circle $x^2 + y^2 = a^2$

An equilateral triangle ABC circumscribes the circle with equation $x^2 + y^2 = a^2$. The side BC of the triangle has equation $x = -a$. a) Find the equations of AB and AC. b) Find the equation of ...
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1answer
20 views

How/why does this proportion work?

In this diagram, ΔXYZ is inscribed into the circles. O is the center of the larger circle. OZ=x, altitude XO=x-5, and OY=x-9. ∠XOZ and ∠XOY are both right angles. Using the two similar right triangles ...
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1answer
50 views

Use a vector method to prove that the triangle is isoceles.

If two medians of a triangle are equal then prove by vector method that it is an isosceles $triangle$ This might be a simple question but i could not do it because i don't know any theorems related to ...
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1answer
13 views

Find area of triangle which sides is limited by two functions and the x axis

I'm studying for my math exam and I'm stuck on the following question "A triangle is limited by the x axis and the two functions $y=kx$ och $y=\frac{1}{k}x+k$ where k > 1. Determine the smalest ...
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1answer
28 views

Determine the height of the tree

Dylan is using his clinometer to help him determine the height of a tree. He stands 6 m from the base of the tree and takes the measurement shown on the clinometer. Then, he measures the height of ...
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2answers
21 views

How to know what type of cross section is it going to be?

A plane intersects a right rectangular pyramid. Producing a cross section. The plane is parallel to the base. What shape is the cross section? I thought it would be triangle cause triangle cut is ...
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0answers
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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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4answers
119 views

What is the approximate area of the shaded region of the given figure?

How do i find the area of the black shaded portion of the circle? I noticed the 4 so i think that's the radius. The formula to find the area $$A=πr^2$$ so I thought of using that to find the area ...
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1answer
24 views

Finding the altitude of an isosceles triangle with base length and angle

So I have triangle ABC where: $AC = BC$; $AB$ is known $\hat C$ (the angle $A\hat CB$) is known I'm trying to find the altitude of said triangle.
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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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1answer
56 views

Prove for a area relationship in a pedal triangle

Let $\triangle ABC$ an acute triangle and call $K,L, M$ the orthogonal projections of $A,C$ and $B$ on the opposing sides. Prove: $A_{\triangle KLM} = 2 A_{\triangle ABC}\cdot \cos \hat A ...
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2answers
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How to solve this exercise

given: $\triangle ABC$ $P=20$ Note: P is perimeter $\cos \alpha = -\frac{1}{3}$ $\cos \beta = \frac{7}{9}$ Find the sides of the triangle I'm totally lost on this one. I have no idea from where to ...
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2answers
61 views

Calculation of the Coordinates of the Intersection Point of the Altitude and the Base of a Triangle

Given any triangle $\Delta ABC$, what are the coordinates of the point $D$, along the line $\overline{BC}$, such that $\overline{AD}$ is perpendicular to $\overline{BC}$? For example, given the ...
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2answers
38 views

Isosceles Triangles in Hilbert Spaces and Metric Spaces generally

In what types of metric spaces $\langle X, d \rangle$ is it possible to do the following? Task: For any two points $x, y \in X$ such that $d(x,y) \leq 2\epsilon$, find a third point $z$ such that ...
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1answer
49 views

Find the area of a particular pentagon associated to a given triangle

Let $ABC$ be a triangle with base $AB$. Let $D$ be the midpoint of $AB$ and $P$ be the midpoint of $CD$. Extend $AB$ in both direction. Assuming $A$ to be on the left of $B$, let $X$ be a point on ...
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1answer
152 views

Is it impossible to construct an equilateral triangle inside a semicircle?

I have made it in a circle(which is very easy)....but I have been unable to make one inside a semicircle....is it not possible to make equilateral triangle inside a semicircle ?... If yes how can we ...
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7answers
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V.I. Arnold says Russian students can't solve this problem, but American students can — why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it ...
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1answer
32 views

Simple geometry: triangle's segments

I have been given a simple homework assignment from kid in my family and do not know how to help him. I am pretty sure that there problem is unsolvable in its current state and he forgot some ...
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3answers
716 views

Solve this geometry problem without using any trigonometry

Given: $ \triangle ABC $ $AC = BC$ $\angle C = 150$ $AB (base) = \sqrt{12}$ Find the radius of the Circumcircle. I have no idea how to solve this challenge, the answer which is given in the textbook ...
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1answer
55 views

How do I rotate a triangle in a graph? [duplicate]

I am trying to rotate this triangle 45degrees counter-clockwise, how do I do that? A = 4,5 (point of rotation (the one that does not move)) B = 4,1 C = ~2.8,1 what is: A', B', C'? I want points ...
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2answers
64 views

Proving result in inscribed triangles.

ABC is a triangle inscribed in a circle, and E is the mid-point of the arc subtended by BC different from the arc A on which A lies. If through E a diameter ED is drawn, show that $$\angle ...
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0answers
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Geometric inequality: $(\text{sum of distances to vertices})>2(\text{sum of distances to sides})$ [closed]

Let $P$ be an interior point of $\triangle ABC$, and let $A^\prime$, $B^\prime$, $C^\prime$ be the projections of $P$ onto respective edge-lines $\overleftrightarrow{BC}$, $\overleftrightarrow{CA}$, ...
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1answer
36 views

Finding vertices of a triangle using complex numbers

The question wants me to find the coordinates of the other two vertices B and C. What I did was that I converted A into complex coordinates, so 4-i. Then I want to find vertex B, and I know that ...
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0answers
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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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3answers
97 views

find the maximum possible area of $\triangle{ABC}$

Let $ABC$ be of triangle with $\angle BAC = 60^\circ$ . Let $P$ be a point in its interior so that $PA=1, PB=2$ and $PC=3$. Find the maximum area of triangle $ABC$. I took reflection of point $P$ ...
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1answer
34 views

Similar spherical triangles are congruent

I look at the following exercise of A. Pressley: Show that similar spherical triangles are congruent. I have no clue how to show it. Can you give an idea how I can do that? In the book ...
2
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1answer
54 views

Prove that given $a,b,c > 0$, it is possible to construct a triangle with sides of length $a,b,c$ if and only if $pa^2+qb^2 > pqc^2$

Prove that given $a,b,c > 0$, it is possible to construct a triangle with sides of length $a,b,c$ if and only if $pa^2+qb^2 > pqc^2$ for any $p,q$ with $p+q = 1$. Should I prove this using ...
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2answers
77 views

Is my definition of a triangle center function “equivalent” to the usual definition?

There's a definition of triangle center function already in existence, but I don't really understand it. Anyway, here's my attempt at defining this concept using ideas I'm more comfortable with. ...
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2answers
69 views

Question on circle and equilateral triangles [duplicate]

Let $ABC$ be a triangle. Let $T$ be its circumcircle and let $I$ be its incenter. Let the internal bisectors of $A,B,C$ meet $T$ at $A',B',C'$ respectively. Let $B'C'$ intersect $AA'$ at $P$ and $AC$ ...
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2answers
21 views

Find Area of Similar Right Triangle

Need help to approach following from GRE study guide Here is what I have so far $area = 1/2 bh$ $area CDE = 1/2 bh = 42 = 21 bh$ $AD = 3CD$ Honestly, I'm not sure how to approach. Please give ...
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1answer
21 views

Rotation of Rectangle Based on a Triangle in 3D Space

I am trying to transform a rectangle centered at the origin and dimensions of $(\| P_2 - P_1 \|, 0, \| \mathbf{V_P} \|)$ to a triangle in 3D space with points $P_0$, $P_1$, and $P_2$ where ...
0
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2answers
94 views

If $\sin(A) = \cos(A)$ find $2 \tan^2 A - 2 \sec^2 A + 5$ [closed]

This is my first try to trigonometry, I have solved 100s of questions but this one always comes incorrect! Can anyone help me? Question: If $\sin A = \cos A$, find the value of: $2 \tan^2 A - 2 ...
4
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3answers
64 views

Proving $\cot { A+\cot { B+\cot { C=\frac { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }{ 4K } } } } $ [closed]

For any acute $\triangle ABC$, prove that $\cot { A+\cot { B+\cot { C=\frac { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }{ 4K } } } } $, where $K$ is the area of $\triangle ABC$. Unfortunately I'm ...
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1answer
57 views

Finding and proving similar triangles

ABC is a triangle with AB shorter than side AC. The angle bisector of ∠A intersect BC at D. Given that point E is on the median that's drawn from A, so that BE⊥AD, how do I show that DE||AB? I tried ...
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3answers
84 views

How to plot a triangle, given three side lengths?

I want to plot a triangle, given side lengths $a$, $b$, and $c$. I can plot point $A$—opposite side $a$—at the origin $(A_x = 0,\ A_y = 0)$. I can plot point $B$—opposite side $b$—along the $x$-axis ...
3
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1answer
129 views

Olympiad Trigonometric Inequality

Let $R$ and $r$ be the circumradius and inradius of $\triangle ABC$. Prove that $$\frac { \cos { A } }{ { \sin }^{ 2 }A } +\frac { \cos { B } }{ { \sin }^{ 2 }B } +\frac { \cos { C } }{ { ...
1
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0answers
46 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| = ...
4
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1answer
29 views

How to triangulate from a Voronoï diagram?

I computed a Voronoï diagram from a set of point (with Boost.polygon). I try to find a Delaunay triangulation, connecting each cell center for each Voronoï edge, but I miss some edges. In the ...