For questions about properties and applications of triangles

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How many ways are there to break up the regular 9-gon into triangles by diagonals?

How many ways are there to break up the regular 9-gon into triangles by diagonals? UPD Guaranteed to be convex - yes. Intersecting "diagonals" be allowed - yes. 2nd UPD It is task for ...
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2answers
19 views

Right angled triangle log

If $a,b$ and $c$($c$ is the hypotenuse) are sides of a right triangle then prove $$(\log_{c+b}a)+(\log_{c-b}a)=2(\log_{c+b} a )\cdot(\log_{c-b}a)$$ The bases are different so can't quite figure out ...
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3answers
20 views

Calculate sides of right triangle with hypotenuse and area or perimeter

I'm trying to find if it is possible to find the lengths of the base and height of a right triangle with only the hypotenuse and the area (or the perimeter) of the triangle. I would have just figured ...
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3answers
38 views

Finding the length of a side of a triangle

I just took the SAT and was wondering if there is any way to find out the length of a side of triangle when you know the three angles and the area of the triangle.
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1answer
10 views

Vectors: right triangle, two vertex known and a direction vector parallel to unknown point

The endpoints of the hypotenuse of a right triangle ABC are A(-10,10,9) and B(14,0,-4). The point C lies on the line that passes through the point A and is parallel to the vector 2i-2j-k. Determine ...
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2answers
68 views

Find an Angle of a Right Triangle Without Trigonometric Functions

I have a right triangle triangle. I know the length of the hypotenuse (H) and one adjacent side (A). I would like to find the angle between the A and the H without using $\arccos(A/H)$. I would like ...
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0answers
15 views

Calculating pairwise distance of two N-dimensional vectors given their length and angle

I am not a mathematician, so apologies in advance for any nomenclature blasphemy. Given the magnitudes of two vectors $b$ and $c$ and the angle between them $A$, I can calculate their distance in 2-D ...
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1answer
20 views

construct a triangle given n items

If $n$ number of inputs are given, then how can I find the number of levels triangle will have. e.g. If $10$ elements are given, there will be $4$ level triangle. If $21$ elements are given, there ...
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2answers
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Trigonometric inequality in an obtuse triangle

Let $ABC$ be an obtuse triangle with $A$ the obtuse angle. I conjecture that the following inequality is true $$\sin B + \sin C \le |\tan A|.$$ Show that it holds or give a counterexample.
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5answers
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Naive approach to Pythagoras

The following has occupied me while learning about $a^2+b^2=c^2$, I then forgot about all that and recently (40yrs after) came across that again - and am still unable to understand. But today my next ...
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8 Angles Question: What are solutions with all angles rational multiples of pi?

I don't know how to draw a picture, maybe someone can help. Consider a convex quadrilateral $ABCD$. The 8 angles I'm referring to are the angles made between the diagonals and the edges. Explicitly: ...
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Proof of equilateral triangle given angles

Let's say we start with a scalene triangle ABC, with no given angle measures or side lengths: Then, we add 3 Isosceles triangles adjacent to this one, given that they have angle measures ...
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2answers
33 views

If in a triangle $ABC$, $a\cos A=b\cos B$, then the triangle is a/an

The options are:- (A)equilateral (B)right angled (C)isosceles (D)either isosceles or right angled Now I took examples to get to the answer but it was wrong. The answer is (D) but I got (C). To check ...
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1answer
38 views

Use calculus to derive area of circle using n triangles

This is a homework question I am struggling with... Let $n$ be a positive integer, and cut the circle into $n$ equal sectors. In each sector there is an isosceles triangle formed where the edges of ...
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1answer
19 views

Perimeter of equilateral triangle from its area

In an exercise, I have to answer the perimeter of a equilateral triangle knowing that its area is $$\sqrt{3}$$ How can I achieve it? I tried inventing equations, but all dead ends. Please explain.
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1answer
16 views

Pendulum tension force

I realize this is physics related, although the problem is really about math so I thought it would be a good fit for this site. My illustration is supposed to depict a pendulum and the forces ...
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1answer
12 views

If a quadratic form $f$ takes the minimum on a triangle in a vertex, what can I say about min of $f$ on edges of a subdivision?

Let $f(x)=x^2+y^2$ be the Euclidean square-norm and $A,B,C\in\mathbb{R}^2$ be vertices of a triangle $\Delta$ such that $f$ takes the maximum on $\Delta$ in $C$, the minimum in $A$ and takes the ...
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1answer
178 views

Combinatorics - Integer sided triangles with integer median

The original problem states: "Given a number N, how many integer-sided triangles $(a,b,c)$ with an integer median $m_{c}$ exist, provided that $a \leq b \leq c \leq N$?". I've managed to get it down ...
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1answer
59 views

Tripartite n+1-regular graph containing a triangle

Suppose a tripartite, $(n+1)$-regular graph. Each one of its $3$ parts $(A,B,C)$ contains $n$ nodes. Show that the graph contains a triangle. I think the fact that it is $n+1$ and not $n$ plays an ...
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1answer
57 views

Interpretation of median length for an invalid triangle

Background: My very first and naive take on the Project Euler problem 513 went wrong, as I counted also triples violating the triangle inequality. Many formulas return an invalid result for an ...
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1answer
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Proof of Compound Angle from Ptolemy's Theorem

I have a query regarding a proof I'm reading on the additive Sine compound angle formula, which uses Ptolemy's theorem. http://www.cut-the-knot.org/proofs/sine_cosine.shtml I'm looking at the ...
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1answer
37 views

Solving triangle

If side $a$ is known and the angles are given as functions of two variables (let's call them $x$ and $y$), what is the easiest way to find $y$ as a function of $x$. To make things easier, let one of ...
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1answer
59 views

finding angle value inside this triangle

I need a method to calculate the angle X in the image below, I know its value (30 degree) but how ?!! thank you.
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1answer
25 views

Given sides and a bisection, find angles in a triangle

Consider a triangle $ABC$ where the angle $A$ is $60^{\circ}$. Draw its bisection intersecting $BC$ at $D$. Let $AB = x$, $BD = y$ and $AC=x+y$, $\angle ABC = \alpha$ and $\angle ACB = \beta$. Find ...
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1answer
37 views

“Reverse engineering” of a geometric illustration

The following enigmatic illustration can be found here, unfortunately without any accompanied comment or short description: Can you deduce its meaning? What was the way it was constructed?
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Trigonometry confusion with triangle in weird question

I was wondering how do you get x from the triangle below:
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0answers
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Conditions for point lying inside triangle formed by three complex numbers.

The question states $z_1,z_2,z_3$ are three non-collinear complex numbers such that $$z=\frac{lz_1+mz_2+nz_3}{l+m+n}$$ lies inside the triangle formed by $z_1,z_2,z_3$. If $l,m,n$ are the ...
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0answers
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How to find local extrema of f (p) give us area of triangle A1B1C1

For a right triangle ABC ( angle C = 90) on the rights height CC1 is chosen point P and consider the triangle A1B1C1 (A1 = AP cross BC, B1 = BP cross AC), if p is distance from point P to AB, to find ...
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4answers
46 views

Find the type of triangle from equation.

In triangle $ABC$, the angle($BAC$) is a root of the equation $$\sqrt{3}\cos x + \sin x = \frac{1}{2}.$$ Then the triangle $ABC$ is a) obtuse angled b) right angled c) acute angled but not ...
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5answers
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Find third point to make isosceles triangle with a specific area

Using points A(1,2) and B(-2,-2), find a third point, with a positive y-value, that makes ABC an isosceles triangle with area 10 units${^2}$. I have found AB to be 5 and used this as $r^2$ below.. ...
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1answer
35 views

Pascal's triangle

I was out sick for a while (2 weeks) and just got back and now we are doing whatever this is! Can someone explain to me what this is or show me a video on how to do it? "Use Pascal's triangle and the ...
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1answer
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finding the value of a node in Pascal’s (a.k.a Yanghui's) triangle [closed]

Image the Pascal Triangle is on an x-y cartesian plane. so that the values of the nodes, by location are ...
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1answer
23 views

Find length of side of a triangle.

Let $ABC$ be a right angled triangle with $BC = 3, AC = 4$. Let $D$ be a point in the hypotenuse $AB$ such that $\angle{BCD} = 30^\circ$. Find the length of $CD$. I found $AB = 5$. How do we find ...
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3answers
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Find circle radius by given triangle inside

So the triangle inside the circle: $AB = 9$cm $CB = 6$cm $CH = 5$cm I think solving this problem involves similar triangles. Thanks in advance, I'd like to have a solution suitable for 9th ...
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How prove $S_{ABC}S_{XYZ}\ge S_{MNP}^2$ for an acute-angled triangle and $M, N, P$ are points from the segments $AB, BC, CA$ respectively

Let $ABC$ is an acute-angled triangle and $M, N, P$ are points from the segments $AB, BC, CA$ respectively. Let $CM\cup NP=X, AN\cup MP=Y, BP\cup NM=Z$. How prove $S_{ABC}S_{XYZ}\ge S_{MNP}^2$? ...
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1answer
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Bounding inradius, given circumradius.

The problem in my book is as follow. In a $\Delta ABC$ , if $r=r_2+r_3-r_1$ and $\angle A >\dfrac{\pi}{3}$ , then the range of $\dfrac{s}{a}$ is equal to: (Here $r_i $ are exradii) I used ...
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1answer
23 views

Finding coordinates of the third point of a triangle from given?

In ABC triangle we know the coordinates of A and B vertices. We also know lengths of 2 edges shown in the picture and the third edge is calculatable. What is the most efficient functon to find x3 and ...
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What is the isotomic conjugate version of the six point circle of isogonal conjugates?

As it is well known, the pedal triangles of a pair of isogonal conjugates in a triangle share a circumcircle. This is a nice theorem, but is there an analogous version of it for a pair of isotomic ...
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2answers
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Find distance between two poles.

2 poles, AB of length 2 metres and CD of length 20 metres are erected vertically with bases at B and D. The two poles are at a distance not less than twenty metres. It is observed that tan(angle(ACB)) ...
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3answers
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Proving $ \frac{1}{c} = \frac{1}{a} + \frac{1}{b}$ in a geometric context

Prove or disprove $$ \frac{1}{c} = \frac{1}{a} + \frac{1}{b}. $$ I have no idea where to start, but it must be a simple proof. Trivia. This fact was used for determination of resistance of two ...
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3answers
583 views

In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. Why?

In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. A statement from the trigonometry section of Simmons' Precalculus in a nutshell. Please ...
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1answer
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Calculate isocele triangle dimensions from angles [closed]

maybe simple but I'm wondering how to dertermine dimensions of an isocele triangle from its given angles and a given height value. Any idea ? Thank you EDIT : this is what I may do : Lets say my ...
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1answer
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How to get a Right Triangle's points' coordination in the space?

I have a Right Triangle with equal legs of 1 unit long rotated on 3 individual angles in the space like in the picture below: As could be seen in the picture, the input I have are the angles 'a' ...
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2answers
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Prove the centroid coordinate formula

How to proof that the coordinate of the centroid of a triangle ABC is given by $\frac{A+B+C}{3}$ using vectors?
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Cabri 3D - Rotating a triangle

I'm given the exercise, in Cabri 3D, to rotate the triangle T around the axis AB and lead it via the triangle To to the triangle T'. I tried to rotate the triangle T around a fixed point and then ...
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2answers
27 views

Layer on which ball belongs in tetrahedron

What is the most computationally efficient way to find the layer on which a ball (i) belongs when arranged in a tetrahedron or 3 dimensional triangle with a triangular base. The ball on the top layer ...
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0answers
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trigonometry - find coordinates of inner triangle after rotation

here is my situation: I have a rectangle I'm rotating 30 degrees counterclockwise, how could I use trig to get the 3 vertices (corners) and lengths of the purple triangle sides and hypotenuse ...
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3answers
220 views

Finding the area of the 4th triangle, given the areas of the other 3, and all the 4 form a rectangle

In one of my tutorial classes, when I was studdying in 9th class (I am in 10th now), our tutor gave us a problem saying it’s a difficult one, and to him, it was incomplete. This is that problem: ...
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Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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1answer
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Find circumcenter when distance between ABC points of triangle with two points's ratio given

The complete problem is: I am having three points A,B,C whose ratio of the distances from points (1,0) and (-1,0) is 1:3 each. Then I need the coordinates of the circumcenter of the triangle formed ...