For questions about properties and applications of triangles

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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 ...
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1answer
45 views

Given 3 points and there distances from eachother find a fourth point equidistant to the 3.

This question can also be asked: given a triangle, and its dimensions, whose vertices lie on the edge of a circle find the radius of the circle. I am not actually sure if there is enough information ...
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1answer
291 views

Conclusion from trigonometric identity

Let $\alpha$ and $\beta$ be angles in triangle, i.e $\alpha, \beta \in \left(0,\pi\right)$ can we conclude that $\alpha = \beta$ if the following statement is true: $$\left(\frac{\sin \alpha}{\sin ...
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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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2answers
136 views

Finding the length of the opposite and adjacent sides of a triangle

I am writing a small game in javascript. It's been a while since I have done any basic maths and I can't get some of my positioning to work properly. Apologies if this question is too simple, but I ...
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1answer
15 views

Position Value in a Triangle

I have a triangle. Each of the 3 corners is assigned a different value. Lets say corner 1 is 100, corner 2 is 200, and corner 3 is 300. I want to pick a coordinate in the middle of the triangle and ...
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1answer
255 views

What is the maximum number of triangles in a planar graph with n vertices?

The answer is obvious for small numbers of nodes: $$n<3: 0\\ n=3: 1\\ n=4: 3\\ n=5: 5 (see below)$$
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2answers
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when to use sine vs cosine vs tangent

I'm a little confused about how you choose to use either sine or cosine or tangent over the others. Are they interchangeable given the same information you have about a right triangle? What are the ...
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1answer
144 views

Area of triangle inside triangle

In triangle $ABC$ we choose 3 points $D,E,F$, such that $\overline{AD} = \frac 13 \overline{AB}, \overline{BE} = \frac 13 \overline{BC}, \overline{CF} = \frac 13 \overline{CA}$. Draw segments ...
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2answers
76 views

Show that if dist(P, ℓ1) = dist(P, ℓ2) then P is on the bisector of the angle

In the image above line ℓ1 refers to $AQ$ and the ℓ2 refers to $AR$. It is given that dist(P, ℓ1) = dist(P, ℓ2). I think I have to show that $PA$ is the bisector of $\angle A$ but I don't know where ...
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How do you find the height of a triangle given $3$ angles and the base side? Image given.

This question has me absolutely stumped. This is the image of the question, how can I work out $x$? I've been doing a variety of attempts but I just cant get it.
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1answer
480 views

How to solve bearing of oblique triangle

I'm having a hard time finding the solution of the bearing given in our example. Our Example: Suppose there's a triangle with points named A,B, and C. Point A is named Bacoor. Point B is named San ...
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1answer
105 views

Calculate side of a triangle with known base and median

Is it possible to calculate the size a of the triangle where t is a median? If so, how?
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3answers
168 views

Guessing the nature of a triangle if one angle is 60 degrees

In triangle ABC, AB=12cm, angle B=60 degrees, the perpendicular from A to BC meets it at D. The bisector of angle ABC meets AD at E. Then E divides AD in the ratio $3:1,6:1,1:1,2:1$? I assumed it ...
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1answer
44 views

Distance to the point of intersection of two altitudes in a triangle

Problem $P,Q,R$ are the points $(4,2)$, $(2,1)$ and $(6,-3)$. Also, $PS$ and $QT$ are altitudes in this triangle. A) Find the equations of PS and QT My answer: PS: $y-x-2=0$, QT: $5y-2x-1=0$ ...
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1answer
421 views

How to identify opposite and adjacent parts of right triangle

If you have a right triangle and both the opposite and adjacent sides have values of ex.10 or the same value. How do you determine which side is the opposite and which is the adjacent if they are both ...
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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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1answer
174 views

Cartesian equation of a triangle

I am wondering what will be the equation of a triangle with vertices at (1,0), (0,-1), and (2,-1)? I really appreciate your quick responses on this! Shah
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2answers
18 views

angle in triangle of pre-known measure

I'm facing a problem. We have lengths of 3 segments. How to see if the triangle built of our 3 segments has a specific angle, for example 60°?
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1answer
320 views

Prove vertex-orthocenter distance is twice of side_midpoint-circumcenter distance

$AL$ , $BM$ , $CN$ are altitudes , $TD$ , $RF$ , $ES$ are perpendicular bisectors of sides. How to prove $AQ = 2PD$ ? By similar triangles $\triangle AQN \sim \triangle CQL$ and $ \triangle CQL \sim ...
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2answers
108 views

Triangle incenter relation

Let $ABC$ be a triangle in which $AB = AC$ and let $I$ be its in-centre. Suppose $BC = AB + AI$. Find $∠BAC$. I do not see how to start even, please help.
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2answers
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When is $Ar(APD)=Ar(ABCD)$?

This question arose while I was answering this question, (we need to show $Ar(\Delta APD)=Ar(ABCD)$). First the original question: $ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets ...
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0answers
214 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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2answers
117 views

Two parallelograms are equal in area.

I tried this question by constructing a line $PD$ therefore forming two triangles $ADP$ and QDP but couldn't establish the congruency relation between the triangles. My approach was that if I have ...
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7answers
482 views

Area of a triangle

The following problem in elementary geometry was proposed to me. As a mathematical analyst, I confess that I can't solve it. And I have no idea of what I could do. Here it is: pick a triangle, and ...
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2answers
428 views

Area of a quadrilateral inside right angled triangle

$ABC$ is right angle triangle. $AB=24 cm$, $BC=10 cm$, $AC=26 cm$. Point $D$ on $AC$ (hypotenuse) bisects $AC$ and connects point $E$ on side $AB$ such that $ED$ is perpendicular to $AC$. Side $AC$ is ...
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2answers
76 views

How $\pi$, $3.1415…$ and $180^o$ are adaptive together?!

I planed following to compute the circle's circumference. The circle's circumference finally can computable from: $$\lim_{\alpha\to0}{\frac{360^o}\alpha d} = 2\pi r$$ I don't want to follow above ...
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1answer
163 views

Distances to line passing through the centroid of triangle

Let $p$ be a line that pass through the centroid of a triangle $ABC$. Unless the line pass through one vertex, then $2$ verices are one side of the line, while the third one is on the other side. ...
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2answers
67 views

Question on inscribed equilateral triangle

Question: $ABC$ and $ODE$ are equilateral triangle with $BC || DE$. If $O$ is the center of the circle, then find the ratio $AQ:QC$ So, my thought on this is that, since we are not given the ...
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3answers
574 views

Find radius of a circle which is tangent to three known lines

I need to find the equation for a circle which is tangent to the following three lines: y=0 x=0 y=-x+0.338334 For the last tangent line equation, I know that it is tangent at the point (0.169167, ...
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4answers
235 views

Using the Law of Sines to find all triangles with given values of two sides and an angle

Our teacher skimmed over this and we have homework over it. Textbook is mostly unhelpful. I'm confused on how ambiguous case works, and everything I see online just confuses me more. I'm not quite ...
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1answer
693 views

equality of triangle inequality

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
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1answer
127 views

slice up a slice of a triangle into n areas of equal size

Figure description: The point $(0, 0)$ is in the upper left corner. The coordinate system grows to the lower right corner. The short sides of the big triangle have the same length. I want to slice ...
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2answers
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Formula for sides of a triangle where the Perimeter equals to the Area [duplicate]

I was wondering if there is a formula that could generate the values of the sides of a triangle where his area equals to his perimeter. I only found that if the triangle is equilateral then ...
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1answer
454 views

Angle between a plane defined by three points (x, y, z are unknown) and the horizontal

I am a novice in mathematics, and I have a question: Suppose that I have 3 points in the space: (x,y,z) for these points are not known for me. given that I know the angles a, b and c (c.f. above ...
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1answer
677 views

Proving triangles congruent with circles

I have a proof that looks like the following, not really sure where to start/how to solve. Any help would be appreciated. Given: circle $S$ and circle $T$ intersect at $M$ and $O$. Prove: $\triangle ...
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Cross section with equilateral triangles and integration

Hello guys so I needed help with a problem which is: Let $S$ be the solid with flat base, whose base is the region in the $xy$-plane defined by the curves $y=e^x$, $y=−2$, $x=1$ and $x=3$, and ...
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178 views

Show that the triangle which satisfy the inequality $\frac{\sin^2 A+\sin^2 B+\sin^2 C}{\cos^2 A+\cos^2 B+\cos^2 C}=2$

Show that the triangle which satisfy the inequality $\dfrac{\sin^2 A+\sin^2 B+\sin^2 C}{\cos^2 A+\cos^2 B+\cos^2 C}=2$ is right angled. My work: $\sin^2 A+\sin^2 B+\sin^2 C=2(\cos^2 A+\cos^2 ...
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1answer
150 views

If $A,B,C$ are the angle of a triangle, then show that $\sin A+\sin B-\cos C\le \dfrac3 2$

If $A,B,C$ are the angle of a triangle, then show that $\sin A+\sin B-\cos C\le \dfrac32$ I tried substituting $C=180^\circ-(A+B)$ and got stuck. I also tried using the formula $\sin A+\sin B=2\sin ...
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1answer
41 views

Simple Question on Triangles…

What times the sum of the squares of the sides of a triangle is equal to the sum of the squares of the medians of the triangle.
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1answer
57 views

A question on similar triangles…

In the given figure, AB, EF and CD are parallel lines. Given that EG = 5 cm, GC = 10 cm and DC = 18 cm, then EF = ??
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1answer
66 views

A question on equilateral triangles

A point $D$ is on the side $BC$ of an equilateral $\triangle ABC$ such that $DC = \frac{1}{4} BC$ . Then $AD^2 = ?$ Image I drew... Options are $13 CD^2$, $9 AB^2$, $6 CD^2$, $12 ...
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2answers
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magical isoceles triangle and 13/15 ratio

It seems eerily magical that $\dfrac {13}{15}$ corresponds within $99.926$ percent accuracy to the height of an isoceles triangle the height of isoceles $= \sqrt {.75} = 0.8660254...$ and $\dfrac ...
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1answer
37 views

Geometry question (finding the coordinates of a point)

Find the coordinates of a point on the y-axis which is equidistant from A(1,-3,7) and B(5,7,-5) I understand an isosceles triangle can be formed from these points, and the x,z coordinates of this ...
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0answers
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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 ...
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1answer
91 views

Find the area of the Grayed triangle Given the following Figure

Can you help me find the area of the gray triangle in the given figure. I'm having a hard time finding the base value of the triangle, I've managed to find the sides for the big triangle but not ...
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1answer
1k views

Why is it called an “orthocenter”? What is the orthocircle?

We know that the orthocenter of a triangle is the place where the triangle's three altitudes intersect. But why is it called that? The *in*center - the intersection of the triangle's three angle ...
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1answer
70 views

Find rotation angle of given image

At first: our aim is to find the total transformation of left house to the right house. What I did it first is translating the house with the center to the origin. I already found out that the ...
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4answers
43 views

If $a$ and $b$ are cathets a right triangle whose hypotenuse is $1$ determine the highest value of $2a + b$

Can some one help me out on where to go? If $a$ and $b$ are cathets a right triangle whose hypotenuse is $1$ determine the highest value of $2a + b$ ?
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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 ...