For questions about properties and applications of triangles

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0
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2answers
29 views

Calculate the angle from the given points coordinates.

I'm trying to figure out the way to calculate the a angle value from given coordinates of three points as showed on the illustration below: I know how to ...
0
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0answers
40 views

Geometry Problem relating similarity.

Given a triangle $ABC$ and $D$ be a point on side $AC$ such that $AB=DC$, angle $BAC=60-2x$, angle $DBC=5x$ and angle $BCA=3x$ prove that $x=10$. Source: 150 Nice Geometry Problems - Amir ...
0
votes
1answer
20 views

Radius of circumscribed circle of triangle as function of the sides

Given the length ot the sides $a , b$ and $c$ of $ \triangle ABC$. What is the length of the radius of the circumcribed circle? After some formula substitution I came to the monster formula: $$ ...
3
votes
1answer
42 views

Weighted Interpolation over Triangle

Question: Is there a modification of simple component-wise barycentric-based interpolation of vertex values (such as colors) that accounts for arbitrary positive non-zero weights assigned to these ...
1
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3answers
44 views

If $R$ is the circumradius of $\triangle ABC$, and $\cos A=\frac1{2R}$, $\cos B=\frac1{R}$ and $\cos C= \frac3{2R}$, then is it unique and its area?

Given that $R$ is the circumradius of $\triangle ABC$, and $\cos A=\frac1{2R}$, $\cos B=\frac1{R}$ and $\cos C= \frac3{2R}$. Then would the $\triangle ABC$ be unique? If so how easily we may find its ...
15
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4answers
166 views

A Triangle Determinant

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
0
votes
3answers
38 views

If the hypotenuse is 4 times the height from A, prove that one of the angles is 15 degrees

In a right triangle (with angle CAB = 90°), suppose |BC| = 4|AD| with AD being the height from A to BC. Prove that the angle BCA is 15°. I had a similar problem but with 22.5°. I thought it would be ...
-1
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1answer
27 views

Smallest turning angle

I know it's probably been answered, but my google-fu is failing me today... I have two 2D points; A and B A has a known Heading. The X and Y coordinates are always positive, if that helps at all. ...
1
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1answer
46 views

Given the coordinates of four points on a plane, how can one determine the shape they form?

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
votes
0answers
47 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 ...
1
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1answer
59 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 ...
-2
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1answer
40 views

Triangle Geometry Help! [closed]

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 $\triangle ADE$?
1
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3answers
91 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 ...
2
votes
3answers
65 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.
1
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1answer
46 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 $m^2_a$, $m^2_b$, ...
1
vote
1answer
40 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. ...
1
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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 $x \leq a + b - d$?
0
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1answer
16 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
votes
1answer
59 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
votes
1answer
96 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
votes
1answer
38 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 ...
2
votes
0answers
22 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 ...
0
votes
1answer
25 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 ...
0
votes
2answers
31 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
32 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
votes
1answer
64 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 ...
-1
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0answers
21 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: $$h^2= X^2+y^2$$ ...
15
votes
2answers
513 views

Is ABC an equilateral triangle

In the figure, AD=BE=CF. Moreover, DEF is an equilateral triangle. Must ABC be equilateral?
-2
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1answer
37 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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3answers
33 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
votes
3answers
73 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
votes
1answer
28 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
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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
votes
1answer
30 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 - ...
5
votes
2answers
204 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
53 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
votes
0answers
23 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
111 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 ...
1
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2answers
116 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
41 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 ...
1
vote
0answers
33 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
62 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
votes
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] $1.0$ ...
1
vote
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!
1
vote
2answers
50 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
50 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
votes
2answers
69 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
192 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
vote
1answer
42 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: ...
1
vote
4answers
101 views

Ratio of Areas of Similar Triangles

First step, I can't find the height. How do you find the height?