For questions about properties and applications of triangles

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2answers
56 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 ...
0
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1answer
262 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 ...
0
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1answer
610 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.
0
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1answer
111 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 ...
0
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1answer
14 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 ...
3
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1answer
404 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 ...
5
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1answer
75 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
191 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 ...
1
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1answer
115 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 ...
0
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1answer
39 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 ...
0
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1answer
106 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.
0
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1answer
57 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 ...
2
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1answer
52 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?
1
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2answers
42 views

Trigonometry confusion with triangle in weird question

I was wondering how do you get x from the triangle below:
0
votes
0answers
56 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 ...
3
votes
4answers
66 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 ...
1
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5answers
288 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.. ...
1
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1answer
88 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 ...
0
votes
1answer
40 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 ...
1
vote
3answers
156 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 ...
0
votes
2answers
49 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 ...
1
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1answer
229 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 ...
0
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1answer
73 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 ...
1
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2answers
242 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)) ...
4
votes
7answers
217 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 ...
10
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3answers
5k views

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

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 ...
0
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1answer
19 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' ...
0
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2answers
422 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?
0
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2answers
51 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 ...
5
votes
3answers
375 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: ...
25
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1answer
625 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 ...
1
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1answer
143 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
votes
2answers
539 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 ...
0
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1answer
28 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 ...
1
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1answer
81 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 ...
0
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1answer
62 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 ...
10
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5answers
887 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 ...
0
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2answers
128 views

Sum of the area of infinite similar equilateral triangles

How would I solve for the side depicted in the picture?
1
vote
2answers
148 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
votes
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 ...
0
votes
0answers
28 views

Angle condition for $a^2+c^2=nb^2$

Find a necessary and sufficient angle condition (independent of $a,b,c$ -- see under "what I have got so far" for examples) such that $a^2+c^2=nb^2$ where $n$ is a positive integer. Note: As usual ...
3
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2answers
70 views

In triangle $ABC$, $a^2+c^2=3b^2$

In triangle $ABC$, we have $a=BC$, $b=CA$ and $c=AB$ as usual. What is a necessary and sufficient condition for $a^2+c^2=3b^2$ to hold? I created this problem as a generalization of $a^2+c^2=2b^2$ ...
2
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1answer
51 views

Construct triangle from three points on base and difference in distances to third vertex

Imagine such a triangle: We know the differences in distances: $\overline{OA} - \overline{BO}$ and $\overline{CO} - \overline{BO}$, as well as the distances between the points on the base: ...
2
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1answer
73 views

Similar triangle side lengths given its area and similar triangle side lengths

I've been working through this task in an old textbook and can't figure out where I'm wrong. I suspect my whole approach is wrong. Task says: Given the side lengths of a triangle that are equal to ...
3
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1answer
63 views

Triangle geometry: $BC^2+AC^2=n\cdot AB^2$.

I am looking for information regarding which triangles $ABC$ satisfy $BC^2+AC^2=n\cdot AB^2$ for $n=1,2,3,...$. I'm sure that work has already been done in this area since it is a fairly simple ...
0
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1answer
81 views

In the figure,What is the ratio of $AE:AD$?

In the figure (not drawn to scale), rectangle $ABCD$ is inscribed in the circle with center at $O$.The length of side $AB$ is greater than side $BC$.The ratio of area of the circle to the rectangle ...
0
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1answer
43 views

Simple area and angles of squares and triangels

This is a question APPARANTLY tested on primary 4 and I am in Sec 2,wondering how to do this question....None of my classmates also could finish the question. Question: ABCD and BFGE are squares.AE ...
3
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4answers
194 views

Recurrence relation for right-angled triangles stuck-together

Given the image: and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i , i = 1, 2, 3, · · ·$ can be arbitrarily picked. How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$? I ...
3
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0answers
110 views

Number of triangles created after $n$ folds of a square

My daughter's grade 8 math homework included the following question. We were unable to find an answer, and I think we must have misinterpreted the question, as it seems way too hard. Fold a ...
0
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1answer
35 views

Angles inequality in acute triangle [duplicate]

Let $\alpha$, $\beta$, $\gamma$ be angles of acute triangle. How to prove that $(\tan(\frac{\alpha}{2}))^2 + (\tan(\frac{\beta}{2}))^2 + (\tan(\frac{\gamma}{2}))^2 \ge 1$? Does left side of equation ...