For questions about properties and applications of triangles

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Creating line,triangle or quadrilateral

Actually it is an algorithm problem, however i cannot solve the problem. So, We have 4 points, how can we know what kind of shape(figure) can be drawn ? I want to learn mathematically. If possible i ...
2
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0answers
39 views

Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a ...
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2answers
44 views

Circle in Triangle Help! [on hold]

P is the center of the inscribed circle of right triangle ABC. If AB=4 and AC=2 , find AP.
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1answer
52 views

Area of similar triangle

Suppose that we are given a triangle whose area is known. put a circle C of radius r inside that triangle. How can we find the area of a triangle similar to the first one and whose inscribed circle is ...
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1answer
34 views

Triangle Geometry Help! [on hold]

A square is inscribed in a triangle with $\angle A=30^\circ$. What is the ratio of the area of $\triangle EFC$ to the area of $\triangle ADE$?
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3answers
73 views

Find side of an equilateral triangle inscribed in a rhombus

The lengths of the diagonals of a rhombus are 6 and 8. An equilateral triangle inscribed in this rhombus has one vertex at an end-point of the shorter diagonal and one side parallel to the longer ...
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3answers
62 views

How to determine if a triangle can be drawn with the given points.

Given $3$ points $$(x_1, y_1), (x_2, y_2), (x_3, y_3),$$ how does one determine whether they are vertices of a triangle? Thanks.
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1answer
39 views

Constructing triangle using side length-median relationship

$$\begin{align} m^2_a&=\frac{2b^2+2c^2−a^2}4\\[4pt] m^2_b&=\frac{2c^2+2a^2−b^2}4\\[4pt] m^2_c&=\frac{2a^2+2b^2−c^2}4 \end{align}$$ Solving for $a$, $b$, $c$ in terms of $m^2_a$, $m^2_b$, ...
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1answer
31 views

Relationship between the altitude of an isosceles triangle and segments drawn to the lateral side from a point on the base.

Question :In an isosceles triangle, the sum of the distances from each point of the base to the lateral sides is constant. I've tried a couple of things, but it seems like this statement is not true. ...
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2answers
30 views

Relation of length of a projection of a point to a line

In the given figure, can it be said that $x \leq a + b - d$?
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1answer
15 views

Determine sides of obtuse triangle

I really cannot figure this question out. Can anyone give me a hint please!? Find an integer $a$, for which $a$, $a+1$ and $a+2$ are the lengths of the sides of an obtuse triangle.
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Cut RAIT pieces from a RAIT cake

We baked a cake shaped like a RAIT (= right-angled isosceles triangle). We sprayed a large number ($N$) of raisins arbitrarily over the cake. Now we want to give each of our two children a piece that ...
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1answer
54 views

Disprove the possibility of such a triangle.

The image is not that good, but, consider the following figure to be true without actually constructing it,how can one person find a $fault$ in it. The blue colour represents perpendicular, The ...
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1answer
47 views

How to find coordinates of 3rd vertex of a right angles triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
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1answer
34 views

Find the area of the triangle using $\frac12\|u\| \,\|v-\operatorname{proj}_u(v)\| $

Real stuck on this and I'm sure I went wrong somewhere. Here is the question. Using points $A=(1, -1)$; $B=(2,2)$; $C=(4,0)$ find the area of the triangle. The book states that the way to find ...
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0answers
16 views

Self-tiling tile-sets of order 2

A rep-tile is a geometric shape that can be partitioned to smaller copies of itself. The order of a rep-tile is the number of small copies. E.g., a square is a rep-tile of order 4. The smallest ...
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1answer
17 views

Calculation of third vertices of a triangle given a vector that should be perpendicular to the triangle plane

I have an isosceles triangle (2 sides same length) with vertices O, A and B. OA and OB are the same length. Vertices O and A are known, with O at origin (0,0,0). A known vector V, should be ...
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2answers
30 views

Using Pascal's Triangle for Binomial Expansion.

I'm trying to answer a question using Pascal's triangle to expand binomial functions, and I know how to do it for cases such as (x+1) which is quite simple, but I'm having troubles understanding and ...
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0answers
30 views

Infinite Sum of Areas of Isosceles Triangles Help

Do the areas of the isosceles triangles form a geometric series? I can't figure it out! Triangle $\Delta ABC$ is formed such that $\overline{AC}$ is on the x-axis, $\overline{AB}$ is on the line ...
2
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1answer
61 views

How $\frac{\cos \alpha_1}{\sin \alpha}+\frac{\cos \beta_1}{\sin \beta}+\frac{\cos \gamma_1}{\sin \gamma}\leq\cot \alpha+\cot \beta+\cot \gamma$

Let are any two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$. How prove that $$\frac{\cos \alpha_1}{\sin \alpha} + \frac{\cos \beta_1}{\sin \beta}+ \frac{\cos ...
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0answers
20 views

Calculus, triangle ratio problem.

I have a right triangle. And a leg length, lets call it $X$, changes with time. I would like to know how hypotenuse changed with respect to time. Here is my solution: $$h^2= X^2+y^2$$ ...
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2answers
496 views

Is ABC an equilateral triangle

In the figure, AD=BE=CF. Moreover, DEF is an equilateral triangle. Must ABC be equilateral?
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1answer
34 views

Complex numbers and geometry

There exist two different complex numbers $c_1$ and $c_2$, that together with $2+2i, 5+i$ form the vertices of two equilateral triangles. Find the product $c_1c_2$.
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22 views

Isosceles triangle's angles calculation

How can I calculate angles in an isosceles triangle? I have an isosceles triangle ($a=b=10$). Is there any general formula that can be applied to all isosceles triangle's angles without having to ...
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3answers
29 views

finding median when all three sides are not given

Let ABC be a triangle with AB=3cm, AC=5cm. If AD is a median drawn from the vertex A to the side BC, then which one of the following is correct? a) AD is always greater than 4cm but less than 5cm b) ...
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3answers
68 views

Triangle Congruence

I have found a problem form internet and got stucked trying to proof or disproof it. It says: Given AD=AE ,BF=FC; Proof △ABE≌△ACD Update 1 The @Matrial's solution seems very promising however ...
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1answer
27 views

If a triangle has side lenghts $a,b,c$ where $c$ is the largest prove that its obtuse if $c^2>a^2+b^2$ and acute if $c^2<a^2+b^2$.

I was thinking about this and I cant get to a formal proof. I have a sort of mental image where you draw $a$ and $b$ perpendicular and the $c$ is too small to connect the two endpoints. So the right ...
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1answer
22 views

The amount of unit squares being covered

$L$ and $i$ are integers, $L$ is the length of edge of outermost square and $i$ is the minimum length divided from $L$. And there are cells or unit squares consisting the whole block. There is a ...
2
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1answer
29 views

How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - \cot A)$?

Consider all triangles $ABC$ where $A < B < C \leq \frac{\pi}{2}$. How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - ...
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2answers
155 views

What is the flaw in this proof that all triangles are isosceles?

What is the flaw in this "proof" that all triangles are isosceles? From the linked page: One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least ...
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2answers
52 views

A simple geometry question

Suppose $ABC$ is any triangle and $BE$ is any line from the vertex $B$ to a point $E$ lying inside the segment $AC$. Let $D$ be any point on $BE$. I would like to verify the following: regardless of ...
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0answers
21 views

Create dynamic cities of perspective angle x

I'm creating a tilemap... I found you can create unique building sizes with perspective with six tiles using parallel projection, whose angles are always 45 degrees... this allows you to connect to ...
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5answers
104 views

Are all triangles where “$a^2 = b^2+ c^2$”, right-angled?

For a right angle triangle, you can say that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Does the converse hold, ie. can you also say that, if the square ...
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2answers
78 views

Proving that an equilateral triangle in the plane cannot have vertices on integer lattice points

Thanks for the help! I've written a more detailed proof. The hints were great.
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3answers
33 views

Triangle Inequality on complex numbers

Problem Let $z= x + iy$, then prove that: $$|x| + |y| \le 2 ^{1/2} |z|$$ Progress I've tried to write $|z|$ as $(x^2 + y^2)^{1/2}$, and to make some algebra after this, but I'm really new at ...
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0answers
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How find a triangle ABC minimizing $\frac{\sqrt{1 + 2\cos^2 A}}{\sin B} + \frac{\sqrt{1 + 2\cos^2 B}}{\sin C} + \frac{\sqrt{1 + 2\cos^2 C}}{\sin A}$?

How find in triangle $ABC$ the minimum value of : $$\frac{\sqrt{1 + 2\cos^2 A}}{\sin B} + \frac{\sqrt{1 + 2\cos^2 B}}{\sin C} + \frac{\sqrt{1 + 2\cos^2 C}}{\sin A}\text{ ?}$$
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3answers
56 views

Right-Angled Isosceles Triangle covering puzzle

Consider a RAIT (right-angled isosceles triangle), from which we remove a RAIT smaller than half its area by a cut perpendicular to the hypotenuse, like this: How many RAITs are required to cover ...
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7answers
2k views

Can one deduce whether a given quantity is possible as the area of a triangle when supplied with the length of two of its sides?

Recently I have found a question like following: In triangle $ABC$, $AB=AC=2$. Which of the following could be the area of triangle $ABC$? Indicate all possible areas: [A] $0.5$ [B] $1.0$ ...
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2answers
58 views

Trigonometry and triangle proof

Question: Prove that in an acute angle triangle ABC: $$\tan A\tan B +\tan A \tan C + \tan B \tan C \geq 9$$ I have no idea where to even begin this question. Please help me!
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2answers
41 views

Prove triangles formed by two midpoints and an altitude are congruent

Triangle ABC has altitude BH. M is the midpoint of AB, and N is the midpoint of CB. Prove triangle MBN is congruent to triangle MHN. Can we say that MN bisects BH? If so, why? If MN bisects BH (at ...
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1answer
46 views

Prove triangle made from two altitudes and midpoint is isosceles

In triangle ABC, AH and BK are altitudes. M is the midpoint of AB. Prove that triangle MHK is isosceles. All I can see is that the angles formed where the altitudes intersect are equal, and since ...
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2answers
68 views

In any triangle ABC, the expression (a + b + c) (a + b - c) (b + c - a) (c + a - b)$ is equal to

In any triangle ABC, give an equivalence to the expression $$(a + b + c) (a + b - c) (b + c - a) (c + a - b)$$ Can somebody help me? Note that ...
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4answers
189 views

Construction of a triangle

I need to construct a triangle with given information: $c = 6$, $h = 4$ and $\alpha - \beta = 30º$. I'll put approximate result for any clarification.
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1answer
39 views

Calculate PQ if AC = 20

I need to calculate PQ knowing that AC = 20. This is what I got so far: If I call the point between P and A, "M" and If I call the angle: $$\measuredangle{QPB} = y$$ Then: ...
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4answers
98 views

Ratio of Areas of Similar Triangles

First step, I can't find the height. How do you find the height?
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2answers
51 views

Find the value of $a$.

please help I'm lost on what numbers to add or what formula to use
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1answer
37 views

Fourier transform of a triangular pulse

So I've been practicing some fourier transform questions and stumbled on this one; To start off, i defined the fourier transform for this function by taking integral from -tau to 0 and 0 to tau as ...
2
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1answer
32 views

Do the medians (or other cevians) form all the triangles?

I want to know whether set of medians of all triangles, or some other class of cevians, can form the set of all the triangles? For example, in the case of altitudes, $(4,7,10)$ is an counterexample. ...
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1answer
21 views

Translate line vertically and calculate intersection on circle

Let's say I have a line extending from the center of the circle at a 45° angle. If I were to translate that line up 212.132 units, how would I calculate the intersection between the translated ...
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1answer
33 views

Geometry, Mensuration

If the diagonal BC passes through center of the circle, then the area of the shaded region in the given figure is \begin{align*} a)\quad &\dfrac{a^2}{2(3-\pi)}\\ b) \quad ...