For questions about properties and applications of triangles

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Triangle and Maxium value

Given any triangle ABC with $a \ge b \ge c$ such that $\frac{a^3+b^3+c^3}{\sin^3(A)+\sin^3(B)+\sin^3(C)}=7$, what is the maximum value of $a$?
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2answers
47 views

A problem of a triangle

Consider a triangle $ABC$ where the median $CM$ is perpendicular to the angle bisector $AL$ and their ratio is $ \sqrt2 : 1 $. The question is to find $\cos A$. Hints? Btw, I do know that the ...
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1answer
94 views

Angle between two rectangles rotated around a point with a gap inbetween

I am trying to find the angle between two rectangles when there is a known gap between them. See this diagram: I have simplified the problem into three triangles, two of which are the same. Here ...
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1answer
73 views

Möbius Transformation of Triangles

I understand that Möbius transformations are angle preserving transformations. Knowing this, my professor asked us to think about how the image of equilateral triangle is not an equilateral triangle ...
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1answer
63 views

Number of ways to form isosceles triangle by picking points on a circle

Given a circle with 24 evenly spaced points, how would you find the number of possible isosceles triangles (which includes equilateral) that can by drawn using the points? My attempt was to say that ...
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1answer
93 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
3
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1answer
476 views

Solving circle's radius only knowing angle & lengths of external triangle OR solving for sides of a triangle partial side lengths

Is this possible? Given that I know the length of Y and Z and the angle of X can I figure out the radius A? If I can't without more information, I can produce another set of data X Y Z at a ...
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3answers
664 views

Minimum distance between point and face

Given a point in 3D space of the form (x, y, z) and a triangle consisting of 3 vectors (also in the (x, y, z) format), how would I calculate the minimum distance between the point and the face of the ...
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0answers
161 views

Question on Proof of Shoelace Formula

I was looking for a way to prove the shoelace formula when I found this proof: For this clockwise order to make sense, you need a point O inside the polygon so that the angles form $OA_{i}A_{i+1}$ ...
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1answer
47 views

$3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle

I am completely lost in this one $3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle, its area is closest to the which integer?
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2answers
171 views

$m$ be the number of distinct non congruent integer sided triangles each with perimeter $15$

Let $m$ be the number of distinct non congruent integer sided triangles each with perimeter $15$ and $n$ be the number of distinct non congruent integer sided triangles each with perimeter $16$ Then ...
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1answer
221 views

Finding the interior angle between two lines of slopes $m_1$ and $m_2$ from a programming perspective

I have been working on a 2-dimensional object creator program that handles manipulations of arbitrary shapes and calculates collision detection between them. The program allows you to input a shape's ...
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1answer
72 views

Where am I wrong in finding area of this triangle?

I was self-reading Mathematics for Economists by Simon and Blume. On page 815, Section 29.4, he has discussed "Norms on Function Space". And here I am stuck: Let $$f_n = \begin{cases} 2n^2-2n^3x, ...
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1answer
139 views

Area of triangle $OAB$

Question is : Consider a circle of unit radius centered at $O$ in the plane. let $AB$ be a chord which makes an angle $\theta$ with the tangent to the circle at $A$ .find the area of triangle $OAB$ ...
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1answer
86 views

finding the area of triangle

its an 8th standard question . but i m unable to solve it. please help Question " the perimeter of a triangle is 84m . the sides are in the ration 13:14:15 . find the area . " (answer should come ...
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3answers
85 views

Area of a critical Triangle

help me to solve this this problem please: In a triangle $ABC$, $\angle BAC$ = $60\,^{\circ}$,$AB=2AC$.Point P is inside the triangle such that $PA=\sqrt{3}$,$PB=5$. What is the area of triangle $ABC ...
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1answer
84 views

triangle inequality given perimeter and area

Show that the following inequality holds between the perimeter $p$ and the area of the triangle $a$. $$p^2 \ge 12\sqrt3\ a$$
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1answer
103 views

Find length of triangle side

There is a triangle $ABC$ where $|CB|=a$, $|AC|=b$ and medians of these sides intersect at a right angle. Find |AB|. I don't know how to use a right angle in this problem. I have a idea to link a ...
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3answers
155 views

Find out the angle of <ABC

Help me to solve it please.how could it be done.I tried but nothing comes out.Help me please
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1answer
392 views

Korean Math Olympiad 1993 (geometry)

Consider a triangle ABC with BC = $a$, CA = $b$, AB = $c$. Let D be the midpoint of BC and E be the intersection of the bisector of A with BC. The circle through A, D, E meets AC, AB again at F, G ...
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1answer
120 views

Triangle Ratio/Proportions Problem

I would like someone to verify that I am solving this problem correctly. I do not remember the theorem that allows me to make the two halves of the triangle proportional. Because (h1/h2 = h1/h2) ...
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1answer
159 views

Triangle ratio of areas

This is a photo that was originally posted on Google Plus. I would like to know how to solve for S. I started by splitting S into two parts S1 and S2 by drawing a line from A to M. I also know that ...
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2answers
46 views

How does this proof of law of sines determine equal angles?

I was reading over this proof of the law of sines and they say that $\angle CAB = \angle DOB$ because of "basic geometry". I do not get it though, how can you say that the angles are equal?
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2answers
339 views

Equations of triangle sides through medians

This has been bothering me for a while. Given a vertex $A(2;-4)$ and the line equations of two medians ($2x-3y-2=0$ and $5x+3y-12=0$), find the line equations on which the triangle sides are. I've ...
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1answer
130 views

Is it possible to approximate all angles with certain pythagorean triples?

With sticks $a,b$ and $c$ of length $3,4$ and $5$, you able to draw a right (tri)angle. But are also able to construct an angle $\cos\alpha=\frac35, \alpha=\arccos(\frac35)=$$53.13010...^°$. Is it ...
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1answer
73 views

Area of a rhombus

$ABCD$ is a rhombus. We are given the the circumradius of triangles $ABD$ and $ACD$. So how do we compute the area and the side and area of the rhombus? I have tried some properties of the ...
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3answers
9k views

In a right triangle, given slope and length of hypotenuse find length of legs.

Say I have a right triangle. I know the slope and length of $c$, how do I find the length of $a$ and $b$?
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1answer
81 views

Nine-point-circle, midpoint of triangle

ABC is the triangle and M, N are midpoints of AB and AC. Points W, X are on AB, Y, Z are on AC such that WM = MX, ZN = NY. Let T be the intersection of WY and XZ, prove that T lies on the nine point ...
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2answers
1k views

Proof for SSS Congruence?

I'm hoping that someone can provide a method for deducing the commonly known SSS congruence postulate? The postulate states If the three sides of one triangle are pair-wise congruent to the three ...
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0answers
40 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 ...
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1answer
2k views

Ratio of angles in a triangle, given lengths of triangle's sides.

If I have a triangle $\,\triangle ABC,\,$ with sides of lengths $\,AB=6, \;BC=4, \;CA=5,\,$ then what can I know about the ratio of $\,\dfrac{\angle ACB}{\angle BAC}\,$?
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2answers
47 views

Problem with finding “x” in triangle

I have got a problem with finding the x. I think the question isn't true or there should more informations on it.
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1answer
267 views

In Triangle ABC , BM and CN are perpendiculars from points B and C on any line passing through A. If L is the mid-point of BC, prove that ML = NL

I found this question in my textbook and I think this question requires the use of the mid-point theorem. I even tried proving the equality using congruence but couldn't seem to make a headway. I am ...
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1answer
26 views

How to maximize the function

I have a triangle $T=ABC$. I want to calculate $\max (a-b)$, where the the angle $ABC = \beta$, and $|AB|=c$ is fixed (pre-known). My guess is $c\times\cos (\beta)$, but I want to prove it. Let ...
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2answers
637 views

How to calculate the angle between two vectors, defined by 3 points on the earth?

I want to develop a formula to calculate the angle between two vectors. The vectors will be OX and OY (from point O to X , and Y), where the points are defined by their latitude and longitude values. ...
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1answer
276 views

What is a generic triangle?

I think the question speaks for itself. I came across this term in one exercise, but am not sure what it is. The definition my textbook gives is a triangle where the three vertices are free. I am ...
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2answers
76 views

Proof using properties of an isosceles or right-angle triangle

Given a triangle $ABC$ with sides $AB=BC$ and angle$\angle B=100^\circ $, prove that $$a^3 + b^3 = 3a^2b$$ where $a=AB=BC$ and $b=AC$, I have tried to use simultaneously the sine and cosine rules as ...
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1answer
3k views

Find height of a triangle given length of three sides?

How can I find the height of a triangle given the length of all three sides? The only solution I could find was to use Heron's formula to find area then $A=\frac{1}{2}bh$ to find height. Is there an ...
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0answers
78 views

Minimize the perimiter of a triangle with an inscribed circle

A circle touches the two legs of an angle. How can one draw a line that intersects both legs, such that the circle lies within the triangle with as sides the two legs and the drawn line, and such that ...
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0answers
64 views

Probability of a triangle in a circle [duplicate]

I'm confused on my calculations on analytic geometry with probability. Things I learned on these were messed up since I was a newbie on these subjects. Here's my problem: Three points are chosen ...
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1answer
109 views

Inequality in triangle

Let $ABC$ be a triangle and $M$ a point on side $BC$. Denote $\alpha=\angle BAM$, $\beta=\angle CAM$. Is the following inequality true? $$\sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0.$$
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2answers
100 views

Finding the measurement of an angle

I have been stumped on this problem for a couple days now, and I would like some help solving it. Here is the picture that I drew up: ABCD is a regular square. Line FG is a perpendicular bisector ...
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2answers
2k views

What is the maximum area of a square inscribed in an equilateral triangle?

What is the maximum area of a square inscribed in an equilateral triangle? Please post the approach to solve the above question.
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1answer
136 views

Prove, square of quadrilateral is the sum of squares of 4 triangles [duplicate]

Let A1, B1, C1, and D1 - midpoints of the sides AB, BC, CD and DA convex quadrilateral AВСD. Directs AC1, ВD1, CA1 and DВ1 - divide it by 5 quadrilaterals and 4 triangles. Prove that the area of ​​the ...
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1answer
690 views

Determine if projection of 3D point onto plane is within a triangle

In 3D, given three points $P_1$, $P_2$, and $P_3$ spanning a non-degenerate triangle $T$. How to determine if the projection of a point $P$ onto the plane of $T$ lies within $T$?
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5answers
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Is there any equation for triangle?

Like there's an equation of a circle, is there any equation of a triangle? I've been trying to build one and the closest thing I've managed to do is to create an equation of 2 lines and use the $x$ ...
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2answers
263 views

In an equilateral triangle what is sum of distance from vertices to a point inside the triangle?

In an equilateral triangle what is sum of distance from vertices to any arbitrary point inside the triangle? What is the relation between $a$ and $x + y +z$. The special condition is that the ...
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0answers
40 views

Trigonometry: Isosceles Triangle [duplicate]

I saw the following problem on Facebook (figure not drawn to scale): ...
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5answers
115 views

Right Triangles and Altitudes

I am once again stuck on a question about geometry, this problem is about altitudes that crate right triangles: Let there be a triangle that has side lengths of 13, 20, and 21. Given this, find the ...
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1answer
68 views

Finding inradius given the heights

I'm given the heights of a triangle. Find the inradius. I know that inradius is area/semiperimeter. But then?