For questions about properties and applications of triangles

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31 views

Ideal Triangles and Klein Beltrami Disc

I'm trying to prove something with the ideal triangle in hyperbolic geometry and someone told me that the ideal triangle looks like a euclidean triangle inscribed in a circle in the Klein Beltrami ...
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1answer
60 views

Geometry: Perimeter of triangle formed by intersections of tangents

I'm a bit stuck on the question below, and I wondered if anyone out here might be able to help: Construct a circle with a centre in O(0,0) and a radius of 5. Two tangents of the circle intersect in ...
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2answers
85 views

Hyperbolic Ideal Triangle

I have everything pretty much figured out everything but I need help proving the unique point formed by the three perpendiculars in the picture
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1answer
95 views

Finding the radius of excircles from a right angled triangle

Right angled triangles have 3 excircles, I'm struggling to find a formula which gives me the radius of all three excircles, I've been struggling with this for a while. I've done some googling and I ...
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1answer
59 views

Generating a random num from a triangular distribution [duplicate]

http://en.wikipedia.org/wiki/Triangular_distribution#cite_note-1 under "Generating Triangular-distributed random variates" given that U is a number between 0 and 1, what happens if the a, b and c ...
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2answers
44 views

Find the acute angles of this right triangle.

I am having trouble finding the acute angles of this triangle. O is the intersection of the medians of the triangle and $OG = \frac{1}{2}OH$. Any suggestions?
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1answer
111 views

How to prove a triangle similarity problem

If I have a triangle $ABC$ with point $E$ lying on $BC$ and point $D$ lying on $AB$ where $AE$ is the height to $BC$ and $CD$ is the height to $AB$, how can I prove that triangle $ABC$ is similar to ...
10
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1answer
187 views

Number of ways to dissect a square into triangles of equal area

Monsky's theorem states that it is impossible to dissect a square into an odd number of triangles of equal area. If $n$ is an even integer, I am interested in the number of ways of dissecting a ...
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0answers
20 views

Statue and a flag distances

Next to a flagpole is a statue that measures 9m high. The upper end of the flagpole with the bottom of the statue form an angle of 53.13 degrees to the floor, and the upper end of the flagpole to the ...
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3answers
83 views

Why are trig functions defined for the unit circle?

Why did we ever need to define the trig functions of angles greater than 90 degrees or less than 0 degrees? What is the use of applying trig functions to such angles? If we apply the trig functions ...
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1answer
41 views

Finding the angle?

I have two circles which share a radius of R units, and each circle contains the center of the other circle. I found that the area of the segment would be, $\theta$ is the central angle between the ...
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1answer
56 views

Incenter divide ratio

Given a triangle $ABC$ and angle bisectors $BD,CE$ which intersect at $O$ (incenter) . The ratio in which $O$ divides $BD$ is $3:2$ and it divides $CE$ in ratio $1:2$ . Find the ratio in which the ...
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2answers
95 views

The conjecture that no triangle has rational sides, medians and altitudes

I have found a conjecture that there is no triangle whose sides, medians, altitudes and area are all rational. I figure that someone must have already found such a triangle if one existed and yet I ...
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1answer
30 views

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
31 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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4answers
559 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
74 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
14 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
89 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
21 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
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
20 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
156 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
21 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
39 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
47 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
75 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
56 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
31 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
288 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
67 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
145 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
66 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
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 ...
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1answer
65 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
34 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
42 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
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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
38 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
19 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
55 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
89 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
55 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
27 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
95 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
30 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
35 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
66 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
31 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 ...