For questions about triangles

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0
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1answer
13 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.
0
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0answers
14 views

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 ...
0
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1answer
51 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 ...
0
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1answer
31 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 ...
0
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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
16 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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1answer
26 views

Find the area of the triangle [on hold]

Let $P=(0,1,0),\ \ Q=(-1,1,2) \ $ and $ \ R=(2,1,-1)$. The answer is $1.5$. How is this solution found?
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2answers
28 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
58 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
19 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: $$h^2= X^2+y^2$$ ...
14
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2answers
471 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
33 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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0answers
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 ...
1
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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) ...
4
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3answers
63 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 ...
0
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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 ...
0
votes
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
votes
1answer
28 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
137 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 ...
1
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2answers
50 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 ...
0
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0answers
20 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 ...
7
votes
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
65 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
31 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
32 views

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 : $$\frac{\sqrt{1 + 2\cos^2 A}}{\sin B} + \frac{\sqrt{1 + 2\cos^2 B}}{\sin C} + \frac{\sqrt{1 + 2\cos^2 C}}{\sin A}\text{ ?}$$
3
votes
3answers
52 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 ...
16
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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
36 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 ...
1
vote
1answer
44 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 ...
0
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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 ...
7
votes
4answers
182 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.
1
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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
97 views

Ratio of Areas of Similar Triangles

First step, I can't find the height. How do you find the height?
0
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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
1
vote
1answer
31 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
votes
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. ...
0
votes
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 ...
0
votes
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 ...
1
vote
1answer
23 views

$S$, $I$, $O$ are circumcenter, incenter and orthocenter then $SO\ge IO \sqrt2$

Let $S$, $I$ and $O$ be the circumcenter, incenter and orthocenter of $\triangle ABC$ then prove that $SO\ge IO \sqrt2$, or equivalently $SO^2\ge 2IO^2$. I was able to derive an expression for $SO^2$ ...
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0answers
13 views

Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
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2answers
35 views

isosceles and tight triangle

Hi, I was wondering if there is a way to find x with only knowing the length of isosceles triangle and no other piece of information.
2
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1answer
39 views

Sum of inradius of constructed triangle

Let $ABC$ be a triangle with inradius $r$ and circumradius $R$. Let $A′B′C′$ be the triangle for which $A′B′$ is the perpendicular to $OC$ through $C$ and so on. Let $r_1$ be the inradius of $A'BC$, ...
0
votes
2answers
47 views

Maximum Area of a Triangle when 1 Side, Perimeter Known

This is an example of a "quantitative comparison" question the GRE would test. Suppose the following information is known: one side of a triangle has length 12 the perimeter of the triangle is 40 ...
2
votes
2answers
43 views

How to find the area of an isosceles triangle without using trigonometry?

I have an isosceles triangle with equal sides $10$ unit, angle between them is $30^\circ$. I need to be confirmed that the area of this triangle can be found in any method other than using any kind ...
0
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0answers
34 views

Translate vertical movement into radial movement?

I've tried all sorts of things, but I'm no mathematician and I've conceded defeat. So I come here for help. I don't know if I really worded the question correctly since I don't even know what I should ...
4
votes
4answers
82 views

Construction of an equilateral triangle from two equilateral triangles with a shared vertex

Problem Given that $\triangle ABC$ and $\triangle CDE$ are both equilateral triangles. Connect $AE$, $BE$ to get segments, take the midpoint of $BE$ as $O$, connect $AO$ and extend $AO$ to $F$ where ...
2
votes
3answers
53 views

For a triangle $ABC$, $a^2+b^2+c^2=8R^2$ then it is a right triangle?

$ABC$ is a triangle, $a^2+b^2+c^2=8R^2$ then how do we prove it is a right triangle?