For questions about properties and applications of triangles

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1answer
21 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
19 views

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
141 views

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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0answers
20 views

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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0answers
30 views

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
34 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
52 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
22 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
20 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 ...
2
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1answer
212 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
61 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 ...
2
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1answer
84 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
44 views

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
38 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
62 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
28 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
38 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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2answers
27 views

Trigonometry confusion with triangle in weird question

I was wondering how do you get x from the triangle below:
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0answers
24 views

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
16 views

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
48 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
69 views

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
39 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
24 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
62 views

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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0answers
24 views

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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2answers
31 views

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
28 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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0answers
20 views

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
66 views

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
94 views

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
676 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
13 views

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
44 views

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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0answers
112 views

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
29 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
10 views

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
240 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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0answers
289 views

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
64 views

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 ...
4
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2answers
87 views

How do I find the Intersection of two 3D triangles?

I've got a rather complicated geometry problem that I'm trying to solve - how to find the intersection between two triangles in 3D space. I've looked around at other questions and answers on this site ...
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1answer
21 views

Coordinate-geometry curiosity question

How can we draw a triangle give one of its vertex and the orthocentre and circumcentre? I tried to invoke the concept of 9 point circle and tried using the centroid but could not succeed in making ...
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1answer
26 views

Analytic-geometry rotation concept

I am confused how my book comes up with the following formula- Lets consider a Right angled Isoceles triangle with $2$ vertices on hypotenuse given as $(x_1,y_1)$ and $(x_2,y_2)$ Now the 3rd ...
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0answers
28 views

Get Tangential Vector from angular velocity

Good day, I'd like to know how to get the tangential vector with magnitude and direction (resolved through x y components) of a vector, given it's angular velocity. In this case, there is only 1 ...
0
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1answer
42 views

Orthocentre of a triangle

How do we determine the orthocentre of a triangle when the vertices are given as $(0,0),(x_1,y_1),(x_2,y_2)$? In a normal case i would take out the equation of any two perpendicular bisectors, get ...
8
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2answers
150 views

Eritrea's Theorem

According to this newspaper, an Eritrean high school student named Saied Mohammed Ali has discovered a new geometric theorem. Another source seems to say that it's the following: Say you have a ...
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2answers
55 views

Sum of the area of infinite similar equilateral triangles

How would I solve for the side depicted in the picture?
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2answers
119 views

Triangle containing most points from a set

Given a point set in $\mathbb{R}^2$, I need to find a triangle connecting three points of the set that contains the most points of the set. Points that lie on the connecting lines don't count. The ...
10
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5answers
2k views

Can area be irrational?

I'm stuck in a question of my book which says: If in an equilateral triangle the coordinates of two vertices are integral then what can we say about the coordinates of the third? The answer is that ...