For questions about properties and applications of triangles

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Triangle $ABC$, with $r + r_1= r_2 + r_3$

An acute angled triangle $ABC$, if $r + r_1= r_2 + r_3$ and $\angle B< \frac{\pi}{3}$ then which of the following is true: where $r$ is inradius and $r_1,r_2,r_3$ are ex-radii. (1) $ b+2c <2a ...
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1answer
11 views

A Quadrilateral and A Triangle in a Trapizium

In the above diagram, $ABCD$ is a Trapizium with $AD || BC$ and $BC \perp AB$ $AB = 20, \; AD = 6,\; BC = 30$ $M$ is a point on $DC$ such that $[ADMB] = [BMC]$, where $[x]$ denotes the area of $x$. ...
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2answers
50 views

Integrating triangle in a 2D plane

I am interested in integrating $(x^2y+y^2x)$ on the following loop: $(x=1,y=2)\rightarrow(x=2,y=1)\rightarrow(x=3,y=3)\rightarrow(x=1,y=2)$. I know this loop forms a triangle with all three sides ...
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0answers
18 views

Using oblique projection can you always rotate a triangle to look like an equilateral triangle? [duplicate]

Starting with any triangle using oblique projection, can you view any shape triangle from an angle to see it as an equilateral triangle?
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3answers
42 views

How do I compute the angles of a pyramid from the angle between its sides?

I have been given the following problem to solve: In a right pyramid whose base is an equilateral triangle, the angle between 2 side-faces is 70 degrees. Compute the base angle of a side-face. I ...
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0answers
14 views

Angle in triangle [on hold]

We have triangle ABC with point E on side AB and D on side BC. We know that AE=AD and BD=AC. Is it true that angle BAD is equal to ABC and if yes, how to show it?
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3answers
30 views

New coordinates after clockwise rotation of triangle?

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle ...
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1answer
34 views

Find $|CM|$, if $|CA|=a$ and $|CB|=b$. [on hold]

Let $O$ be a center of a circle, circumscribed over $\triangle ABC$. Perpendicular, drown from the point $A$ on the line $CO$, cross the line $CB$ in the point $M$. Find $|CM|$, if $|CA|=a$ and ...
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1answer
33 views

Prove that $MN = \dfrac{|b − c|}{2}$

In triangle $ABC$, point $M$ is the midpoint of $BC$ and $N$ is on the angle bisector of $\angle A$ such that $MN \parallel AB$. Prove that $MN = \dfrac{|b − c|}{2}$. Attempt: I drew it out and ...
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0answers
43 views

In a triangle with sides $a, b,c$ and the relation $a^3=b^3+c^3$, get the angle between $b$ and $c$. [on hold]

In a triangle with sides $a, b,c$ and the relation $a^3=b^3+c^3$, get the angle between $b$ and $c$.
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2answers
41 views

Geometry/ Triangles problem

I have been struggling with this problem, and I think it should be possible to solve but right now I cannot find how. Given two coordinates/points (x1,y1) and (x2,y2) The angle d1 with the ...
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3answers
33 views

Find the ratios of the sides of a triangle

If the perimeter of a the right-angle triangle is six times its smallest side, find the ratios of the three sides. I tried solving it by using the normal area and volume.
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0answers
39 views

Pythagorean theorem question

In an isosceles triangle, the length of each leg is $13$ and the length of the base is $24$. What is the length of the altitude drawn to the base?
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1answer
31 views

Proof concerning isosceles triangles

In the triangle $ABC$ it is $AC = BC$ and $\alpha = \beta$. The points $D$ and $E$ are on the line through $A$ and $B$. Show that the triangle $CDE$ is isosceles. Hey there! Is it sufficient ...
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1answer
24 views

Proof with area segments in a triangle

I have to show that $A M_CS$ and $M_CBS$ have the same area $X$ and that concerning areas $X=Y=Z$ is true. I'm really stuck here, I would appreciate any help or tip...! How can I start here?
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3answers
49 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
37 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
43 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
57 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
31 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
33 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
47 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
44 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
32 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
44 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
47 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
34 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 ...
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2answers
122 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
26 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
19 views

Let $W1 = 0.5$, $W2 = 0.75$, and $\theta = 1$, find two vectors that satisfy $w\cdot x = \theta$.

Let $W1 = 0.5$, $W2 = 0.75$, and $\theta = 1$, find two vectors that satisfy $w\cdot x = \theta$. Can someone please guide me? I know I'm supposed to use $a \cdot b = \|a\| \|b\| \cos( \theta )$.
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0answers
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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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1answer
42 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
20 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
40 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 ...
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2answers
86 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
23 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 ...
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3answers
60 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 ...
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3answers
49 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 ...
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1answer
57 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 $T'$ ...
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2answers
41 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 ...
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3answers
49 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
42 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
29 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
59 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
55 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? ...
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1answer
30 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 ...
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0answers
32 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
41 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 ...
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2answers
32 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 ...