For questions about properties and applications of triangles

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$H$ is an orthocenter of triangle $ABC$

$H$ is an orthocenter of angle $ABC$. Angle $B$ is $60^{\circ}$. Perpendicular bisectors of $AH$ and $CH$ cross line $AC$ at points $A_{1}$ and $C_{1}$. Show that the centre of $A_{1}HC_{1}$'s ...
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2answers
376 views

Area of one of four regions within a rectangle

There is a figure below (a rectangle). You can see different colors depicting different regions of the figure. The labels on the top of a region defines the area of that region. Can you find the ...
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1answer
84 views

Ortocenter and incenter

In triangle $ABC$: $H_{1}$ is a foot of an altitude from side $BC$, $H_{2}$ is a foot of an altitude from side $AC$, $H_{3}$ is a foot of an altitude from side $AB$, $M_{1}$ is midpoint of $BC$, ...
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1answer
103 views

Triangle Inequality Property for the Euclidean Metric

I've read in many of my books that the triangle inequality for a metric space of the Euclidean Metric is defined as: $$d(x,y) \leq d(x,z) + d(z,y)$$ But when I look up the proof, to help me ...
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2answers
56 views

Diophantine Equation Related To Triangles

a,b and c are the sides of a triangle and a, b, c are integers. I need to solve the following Diophantine equation for positive integral values of k. $bc(b+c-a) = k^{2}(a+b+c)$ I think some ...
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2answers
355 views

double integral over an arbitrary triangle

Assume we have an arbitrary triangle ABC in x-y plane and we want to integrate a function $f(x,y)$ over surface of this triangle as shown in fig. 1: We can define another coordination system [x' ...
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2answers
77 views

$ \cos {A} \cos {B} \cos {C} \leq \frac{1}{8} $

In an acute triangle with angles $ A, B $ and $ C $, show that $ \cos {A} \cdot \cos {B} \cdot \cos {C} \leq \dfrac{1}{8} $ I could start a semi-proof by using limits: as $ A \to 0 , \; \cos {A} ...
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1answer
368 views

Dividing a triangle into seventeen equal parts.

I was trying to solve a problem on Pigeonhole principle from Problem Solving Strategies by Arthur Engel. A target has the form of an equilateral triangle with side 2 units. If it is hit ...
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1answer
217 views

Trigonometry: Find the side of a triangle within a triangle

Please help. I found a solution to this problem on yahoo answers but I do not understand the answer. I would use the laws of cosine but I have to be able to answer this without a calculator If AB = ...
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1answer
84 views

How to prove two triangles have the same centroid?

Suppose you have a $\triangle ABC$ and three similar exterior triangles $\triangle BCX$, $\triangle CAY$ and $\triangle ABZ$. How can I prove that the centroids of $\triangle ABC$ and $\triangle XYZ$ ...
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1answer
48 views

$a\in\mathbb{Z}/\mathbb{Z}_{p}$ and Triangles

Can someone please give me a geometric explanation for the following: if you consider the integers modulo a prime as a group under + and plot the set of points $(a,a')$ where ...
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2answers
96 views

What are the ranges of triangle angles?

Lets say, that $\alpha \le \beta \le \gamma$. As shown here, $60 \le \gamma \lt 180$. What are the minimum and maximum values of $\alpha$ and $\beta$? The answer: $$0\lt \alpha \le 60 \\ 0 \lt ...
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1answer
51 views

What is the range of angle in front of longest triangle edge?

What is the minimum and the maximum values of the angle $\gamma$ in front of the longest triangle edge?
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2answers
230 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 ...
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1answer
72 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: $$ ...
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1answer
357 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 ...
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3answers
55 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 ...
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4answers
278 views

A triangle determinant that is always zero

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

If the hypotenuse is $4$ times the height from $A$, prove that one of the angles is $15^\circ$

In a right triangle (with $\angle CAB = 90^\circ$), suppose $|BC| = 4|AD|$ with $AD$ being the height from $A$ to $BC$. Prove that $\angle BCA$ is $15^\circ$. I had a similar problem but with ...
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1answer
41 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. ...
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1answer
396 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 ...
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0answers
55 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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1answer
116 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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3answers
452 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
211 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
110 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
259 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
35 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
27 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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72 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
6k views

How to find coordinates of 3rd vertex of a right angled 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
43 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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1answer
160 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
48 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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1answer
76 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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698 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
72 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
142 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
108 views

Prove two triangles are congruent

I have found a problem form internet and got stucked trying to proof or disproof it. It says: Given $AD=AE$, $BF=FC$, prove $\triangle ABE\cong\triangle ACD$ Update 1 The @Matrial's solution ...
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1answer
38 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
28 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 ...
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1answer
37 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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3answers
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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
60 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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59 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 $45^\circ$ .... this allows you to connect to ...
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5answers
149 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
176 views

The concurrence of angle bisector, median, and altitude in an acute triangle

$ABC$ is an acute triangle. The angle bisector $AD$, the median $BE$ and the altitude $CF$ are concurrent. Prove that angle $A$ is more than $45$ degrees. Here $D,E,F$ are points on $BC,CA,AB$ ...
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672 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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78 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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3answers
201 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 ...