For questions about properties and applications of triangles

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3answers
88 views

Prove a parallelogram inside parallelogram

I have drawn a figure, In parallelogram ABCD, AP is the bisector of angle A CQ is the bisector of angle C Can I prove APCQ is a parallelogram? or it isn't? I first joined AC and now if somehow I ...
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2answers
47 views

I need some help with Geometry. Is this a correct answer to this problem?

Good day, I have a question regarding geometry. I don't know whether my answer is correct because the answer in my book uses a totally different method for solving this particular problem. Here's ...
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3answers
60 views

Distance of centroid to incenter

Suppose there is a right triangle where all side-lengths are integers. The distance from the circumcenter to the centroid of the triangle is 6.5. Find the distance from the centroid to the incenter ...
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2answers
63 views

is there a triangle with sides $2,3,6$?

Is there a triangle with $a=2, b=3, c=6$? (I know there's not because sum of any two sides has to be greater than the third side) How much do we need to extend these sides to get a right triangle ...
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1answer
32 views

Find sides of a right triangle given hypotenuse c and area A (no numbers given)

I've solved couple of these, but I have no idea how to solve it without any numbers provided. I've tried using $A=\frac{ab}{2} \Rightarrow 2A=ab \Rightarrow 4A^2=a^2b^2$ and incorporating ...
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0answers
38 views

Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
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1answer
84 views

A problem with “Crossed Ladders Theorem”

In the following diagram, $AY ||BZ$, $AB$ is base. $M$ is $5$ above $N$ and $N$ is $4$ above $O$. What is the height of the triangle $\Delta AOB$. My Work There is a theorem named Crossed Ladder ...
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3answers
71 views

Inside an not Equilateral Triangle what is the sum of distances from a random point to 3 sides

Given an not Equilateral Triangle with following side sizes: 45,60,75. Find a sum of distances from a random located point inside a triangle to its three sides. Note 1: Viviani's theorem related only ...
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1answer
38 views

Inequality in triangle.

If $a,b,c$ are sides of a triangle prove that- $$\frac a{c+a-b}+\frac b{a+b-c}+\frac c{b+c-a}\geq3$$ I am having problem in approaching the problem as the sides are not mentioned as ...
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1answer
58 views

2011 AMC 12A #13 — Different answers to triangle geometry problem

Triangle ABC has side lengths $AB = 12$, $BC = 24$, and $AC = 18$. The line through the incenter of triangle ABC parallel to $\overline{BC}$ intersects $\overline{AB}$ at M and $\overline{AC}$ at ...
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3answers
62 views

Circle through the circumcentre of a triangle problem

Let ABC be an acute triangle and O it's circumcentre. Let S denote the circle through A,B, O. The lines CA and CB meet S again at P and Q, respectively. Prove that the lines CO and PQ are ...
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1answer
88 views

New SAT Math Section: Pythagorean Theorem on Soccer Fields

So I attempted this problem and I'm very sure I'm doing it right but I keep getting it wrong as my answer choice is not even one of the answer choices listed. There is a picture that goes with the ...
1
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2answers
129 views

Prove $(a^2+b^2+c^2)(a+b-c)(b+c-a)(a+c-b)\leq abc(ab+bc+ac)$

Let $a,b,c$ are $3$ edge of a triangle. Prove $(a^2+b^2+c^2)(a+b-c)(b+c-a)(a+c-b)\leq abc(ab+bc+ac)$. My try: I suppose $c=\min\{a,b,c\}$ but I don't know what next.
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1answer
49 views

Finding a 3rd point in a 3D triangle with known plane, two points and lengths of each side

I have a very similar problem to the below question. right triangle in 3D space, vectors, line intersection? Rather than having the unit vector $A$ I have the lengths $i_2$ to $i_3$ and $i_1$ to ...
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0answers
37 views

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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1answer
75 views

moduli space of triangles

I found an article which seems to be aimed for general audience. I couldn't understand sentences about triangles. The link to the article is the following. ...
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2answers
23 views

Lowest possible value for k for triangle with an integer area

There is a triangle with sides length (9 + k), (39 + k), and (48 + k). The triangle has an area that is an integer. What is the smallest possible value for k? I already tried pythagorean theorem.
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2answers
49 views

Finding an angle between two vectors

I am trying to answer part $d)$ by using my answer to part $c)$. From what I can see, the only possible way to do this is to find the lenght of $AB$ and $OB$, and, using the angle in part $c)$, apply ...
4
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2answers
92 views

Finding segment in a right triangle.

Here is the picture of the question: $ABC$ is a right triangle. $m(CBA)=90^\circ$. $m(BAD)=2m(DAC)=2\alpha$. $D$ is a midpoint of $[BC]$. $E$ is a point on $[AD]$. ...
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1answer
25 views

$3$ Triangles and a quadrilateral

In the following diagram, in $\Delta ABC$, $CD$ and $BE$ are two cevians intersecting it point $O$. Area of $\Delta BOD = 3, \Delta BOC = \Delta COE = 7$. What is the area of $ADOE$. Note: I can't ...
3
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3answers
119 views

How to determine (and explain) the sum of angles without measuring?

Below is a photo of the angles/triangles in which I am working on determining the sum of the angles without measuring. The angles are a,b,c,d,e,f. I understand that angles are formed my ...
2
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3answers
176 views

Find the two other sides in a 15-30-135 triangle

A triangle has angle measures of 15, 30, and 135 degrees. The side opposite the 15 angle is x feet, the side opposite the 30 angle is y feet, and the side opposite the 135 angle is 2 feet. Find x and ...
4
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1answer
61 views

Triangle with same black and white areas

Suppose we have an infinite chessboard with the usual black/white coloring. A triangle $T$ with area $a$ is given with vertices at corners of some cells. Prove that there exists another triangle $T'$ ...
3
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2answers
92 views

Prove $||a| - |b||$ is less than or equal to $|a-b|$

I was given the hint to split it into two cases ($|a| - |b|$ being positive and negative) and then use the triangle inequality. However, since the triangle inequality says that $|a+b|$ is less than or ...
4
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3answers
55 views

Prove the triangle is equilateral

HINTS ONLY please. This is very confusing right off the bat. My guess was that we show the angle $C, M, N$ are all $60^{\text{o}}.$ But I am having difficulty doing as as none of the givens have ...
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0answers
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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1answer
32 views

A trigonometry based triangle problem

In the triangle ABC below, side a is 10 units, and side b is 12 units. cos(angleACB) = 1/5. Find the value of cos(angleCBA). I'm pretty sure that I should use the law of sines, or the law of cosines, ...
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3answers
41 views

geometry perimeter for triangles

I don't get.. The largest side of the triangle (side a) is 10 more units that the smallest side (side b) and the 3rd side of the triangle(side c) is triple the smallest side of the triangle. if the ...
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5answers
68 views

Why $a^{2}+b^{2}\neq c^{2}$ when $a=b=c=1$ doesn't violate Pythagoras' Theorem?

My exercise is this: An equilateral triangle whose side lengths are equal to 1. Observe that in this particular case, $a^{2}+b^{2}\neq c^{2}$. Explain why this doesn't violate the Pythagoras' ...
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3answers
74 views

Common area between Circle and Equilateral triangle [closed]

A circle is drawn with diameter BC of a equilateral triangle ABC. Area of triangle is $\pi - 3$ less than the area of the circle. What is the area of the common region between circle and the triangle? ...
1
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1answer
34 views

Given the lengths of two sides of a triangle, is it possible to obtain (sharp) bounds for the average length of the sides of the triangle?

The title says it all. Given the lengths of two sides of a triangle, is it possible to obtain (sharp) bounds for the average length of the sides of the triangle? Let the lengths of two sides of ...
2
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0answers
33 views

Catalog of triangles

Is there a catalogue of triangles in which one might find for instance the name of the right angle triangle with an angle of approx 35 degrees in which the altitude, median and side bisectors ...
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2answers
55 views

Coordinates of circumcentre of an isosceles triangle in 3D

I have an isosceles triangle in 3D and I need to find the coordinates of the circumcentre of this triangle. I know the coordinates of the three vertices. One method I thought of is to solve equation ...
0
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2answers
55 views

angles in rhombus when an equilateral triangle is inscribed in it [closed]

When one inscribs an equilateral triangle in a rhombus, all the corners are multiples of 30 degrees. I can see this, but I can't proof it. Question: How can I proof that the angle ADC is 120 ...
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0answers
47 views

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 ...
0
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1answer
29 views

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 ...
0
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1answer
22 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 ...
2
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3answers
77 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
28 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
47 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
47 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 ...
0
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4answers
82 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 ...
4
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1answer
49 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
48 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 ...
0
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1answer
30 views

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
29 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 ...
0
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1answer
71 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
21 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 ...
0
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1answer
117 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 ...
0
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1answer
20 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 ...