For questions about properties and applications of triangles

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2answers
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Geometry and property of triangle

$ABC$ is a cyclic triangle and bisector of angle $B\widehat{A}C$, $A\widehat{B}C$ and $A\widehat{C}B$ touches circle at $P$, $Q$ and $R$ respectively then measure of angle $R\widehat{Q}P$ is? The ...
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3answers
53 views

Cover a polygon with polygons

Besides right angled triangles, is there any polygon I could use to cover any given (regular or not) polygon? It's clear that given a triangle, square, hexagon or rectangle you would other options. ...
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1answer
28 views

How to do Vectors with triangles?

This is my homework and I have just started learning it and I don't really quite understand it.
5
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4answers
143 views

In $\triangle ABC$, if $\cos A\cos B\cos C=\frac{1}{3}$, then $\tan A\tan B+\tan B \tan C+\tan C\tan A =\text{???}$

In $\triangle ABC$, if $$\cos A \cos B \cos C=\frac{1}{3}$$ then can we find value of $$\tan A\tan B+\tan B \tan C+\tan C\tan A$$ ? Please give some hint. I am not sure if $\tan A \tan B+\...
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2answers
34 views

Prove that an Equilateral cannot have natural number points

Let $ OAB $ be an equilateral triangle with $O(0, 0),\ A(m, n),\ B(x, y)$, where $m, n \in \mathbb{N}^{\ast}$ and $x, y \in \mathbb{R}_{+}$. Prove that $B$'s coordinates can't be both natural numbers....
2
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1answer
35 views

General Triangles: Area, lengths and angles calculations

I have a question on General Triangles (as in non right angle). I’m trying to create a program that calculates angles and sides based on the user entering Area and some sides length or angle ...
0
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1answer
90 views

Given latitude and longitude, how to find central angle and arc length of spherical triangle?

Lewis and Clark followed several rivers in their trek from what is now Great Falls, Montana, to the Pacific coast. First, they went down the Missouri and Jefferson rivers from Great Falls to Lemhi, ...
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0answers
28 views

Find the length of a triangle

Question is: Find the length of $\text{AO}$ and $\text{BO}$ My work, with the things I know already: Length: $\text{CO}=r$ and $\text{TO}=\frac{r}{4}$ and $\text{TG}=\frac{r}{2}$ and $\...
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0answers
10 views

Sampling from Irwin-Hall distribution using triangular distribution

So I need to sample from the Irwin-Hall distribution using rejection sampling with the triangular distribution. I built 2 functions: The first is d_irwin which receives an $x\in supp(g)$ and the n we ...
0
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0answers
70 views

Calculate the coordinates of two points in an isosceles triangle

In the triangle below, given the point A, angle θ and length d of the two equal sides, how can the points B and C be calculated? Edit:After brainstorming for quite some time, I ended up with a ...
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1answer
90 views

Relationship between incenter and circumcenter

Let ABC be an acute triangle with circumcenter O and incenter I. Points E, M lie on AC and F, N on AB so that BE ⊥ AC, CF ⊥ AB, ∠ABM = ∠CBM and ∠ACN = ∠BCN. Prove that I lies on EF if and only if O ...
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3answers
48 views

Find area of triangle ABC given areas of sub-triangles

The line p is parallel to the the side AB of triangle ABC and splits the sides AC and BC in points D and E, respectively. If the area of triangle ABD is m and the area of triangle AEC is n, find the ...
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2answers
45 views

Finding sides of triangle

Given : $$\triangle ABC$$ $$M \in AB,N \in BC ,P \in AC$$ are the points at which the incircle crosses the triangle $$MN=3\sqrt{10}$$ $$NP=2\sqrt{20}$$ $$PM=10$$ I have to find the sides of the ...
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0answers
45 views

Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem (...
3
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1answer
39 views

Circumcentre and Incentre [closed]

if I is the incentre and S is the circumcenter of ABC prove that angle IAS is half the difference between angle B and angle C
2
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2answers
78 views

Circle inscribed in Equilateral Triangles

The circle inscribed in the triangle $ABC$ touches the sides $BC$ , $CA$ , and $AB$ in the points $A_1,B_1,C_1$ respectively. Similarly the circle inscribed in the triangle $A_1B_1C_1$ touches the ...
2
votes
1answer
48 views

Point which minimizes the squared sum distances to edges of a triangle [duplicate]

I need to find coordinates of a point at which the sum of the squared distances to each of the three lines that form a triangle is minimized. It seemed to me that the point is the triangle's incenter,...
0
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1answer
26 views

X is any point on AB and the median AD of triangle ABC meets XC at Y.Prove that XY/YC = AX/XB

X is any point on AB and the median AD of triangle ABC meets XC at Y.Prove that XY/YC = AX/XB.
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1answer
36 views

Write down the iterated integral which expresses the surface area of z = y 6 cos3 x over the triangle with vertices (-1,1), (1,1), (0,2):

This problem has been done a few times on other sites but there is no work or explanation of the steps taken to get h(x,y). I can understand getting the limits by finding the bounds of the triangle so ...
0
votes
1answer
45 views

Finding angle associated with point inside an equilateral triangle.

$\triangle{ABC}$ is an equilateral triangle. $|AD|=6$. $|BD|=10$. $|CD|=8$. What is $m\angle{CDA}$? First thing comes to mind is Ceva theorem. I used its trigonometric form to reach ...
2
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1answer
24 views

triangle park problem [closed]

We have a park that is triangle. We don't know the shape of the triangle and it can have any triangle shape and lengths. Where should we place a lamp to have light everywhere in the park? my english ...
0
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1answer
31 views

Pythagorean Theorem on Spiral of Theodorus Triangles

I have 1 right triangle of dimensions $\sqrt75$$, 11, 14$. I'd like to know how to quickly obtain the other right triangles with $\sqrt75$ as a leg, and two integers as the hypotenuse and the other ...
0
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1answer
28 views

Get a point by a given point, degree and distance

I have a point $(x_1, y_1)$, an angle $A$ and a distance $D$ How do I get the point $(x, y)$ which is $D$ unit away from $x_1, y_1$ with angle $A$. In the right image, the answer will be $(x_1 - D\...
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2answers
40 views

geometry - prove that you can make new triangle with..

I have a triangle, the length of heights are $i,h,g$. Prove that we can build a new triangle so that the lengths of the sides are: $i^{-1}, g^{-1}, h^{-1}$ (see picture)
2
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2answers
78 views

Joint PDF of two random variables in a triangle

Let the random variables $X$ and $Y$ have a joint PDF which is uniform over the triangle with vertices at $(0, 0), (0, 1 )$ and $(1, 0)$. Find the joint PDF of $X$ and $Y$. So ...
5
votes
2answers
1k views

Determine the third point in right triangle only knowing the coordinates of the other two points

I have a right triangle $ABC$. I am given the coordinates of the two points $A(x_1, y_1)$ and $C(x_2, y_2)$. Given points $A$ and $C$, I want to determine the coordinates of $B$. I know there are two ...
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0answers
31 views

Area of non-spherical triangle on a sphere

This is a followup to the question Area of triangle on a sphere (not spherical triangle) Since it's now almost two years later, I'm making it a new question. The problem is to find the area of ...
-1
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1answer
42 views

Rotate right triangle with perimeter 1 about the hypotenuse [closed]

We rotate every right triangle with perimeter 1 about its hypotenuses. Is it true that we can choose a solid from so obtained solids that has maximum volume? If yes, what's the volume? I guess I ...
0
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1answer
41 views

The ratio of areas of two triangles with the same altitude is equal to the ratio of their bases

How can we prove that the ratio of areas of two triangles of equal altitudes is equal to the ratio of their bases?
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1answer
94 views

Inequality involving circumradii

Let $ABC$ be a triangle and $M$ a point on the side $BC$. Let $R_1$,$R_2$, and $R$ be the circumradii of the triangles $ABM, ACM$, and $ABC$. Show that $\max\{R_1,R_2\} \geq R\cos\frac A 2$.
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2answers
66 views

Three Altitudes of a triangle are concurrent

I have been told that this well known fact can be shown using only Euclid's propositions from books one to three, and cyclic quadrilaterals. I can't figure out how to start, which quadrilateral ...
1
vote
1answer
31 views

Law of Cosine formula that I can't seem to rearrange.

I was in midst of solving a trig problem, and it required using the formula of Law of Cosine. For my case, I had to solve for a specific variable which was $\cos (A)$. Would you show me step-by-step ...
1
vote
1answer
49 views

Equilateral triangle with vertices whose coordinates on the Cartesian plane are integers. Does such a triangle exist? [duplicate]

Can you build an equilateral triangle on a Cartesian plane whose vertices only have integer values as their coordinates? Looking at the simplest example, i.e. a triangle with vertices (0,0), (1,0) ...
0
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1answer
37 views

Geometry: Determining the length of a side of a triangle [closed]

Triangle $ABC$ has all sides of integral length. Vertex $A$ is at $(0,0)$, $B$ lies on the line joining $(0,0)$ and $(3,6)$ and $C$ lies on the line joining $(0,0)$ and $(2,-1)$. Two of the three ...
1
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1answer
37 views

Finding the length of a side of an equilateral triangle

There is a large right isosceles triangle with a hypotenuse length of $24$. Inside the triangle is an equilateral triangle with a vertex on the midpoint of the hypotenuse. If the length of each side ...
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1answer
31 views

Solving for length of an unknown side of a triangle.

I have been given the figure below: Figure (click me). I know that $AD=20-x$ and $m\angle ACD=m\angle BCD$. How can I set up a ratio also knowing that $AC=11$ and $BC=14$ in order to find $x$? ...
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3answers
40 views

Find the measure of a side and an angle.

In the figure, $BG=10$, $AG=13$, $DC=12$, and $m\angle DBC=39^\circ$. Given that $AB=BC$, find $AD$ and $m\angle ABC$. Here is the figure: I am inclined to say that since $\overline{AB}\simeq \...
1
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1answer
40 views

How to calculate triangle coordinates in cartesian plane?

My problem can be describe by following image: I know coordinates of an example P point. Say, they are equal to (8,8). I also ...
0
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4answers
71 views

Properties of Equilateral Triangles in Circles

If there is an equilateral triangle in a circle, would the midpoint of any of the 3 sides be half the radius? e.g if the radius was 6 and at the midpoint of the triangle (call it B) would center to ...
0
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1answer
27 views

Proof formula for triangle: $b^2|BD| +a^2|AD| - c|CD|^2 = c|AD||BD|$

given is a triangle with corners A,B,C and corresponding sides a, b, c. D is a point somewhere between A and B. I have to proof: $b^2|BD| +a^2|AD| - c|CD|^2 = c|AD||BD|$ Unfortunately I have no ...
0
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1answer
37 views

Find the width/height of a triangle given a side length and two lines

I'm a programmer and I came across a math problem in my current project that I can't figure out. My situation looks like this. Everything in black is known or I know how to figure out. ...
6
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0answers
109 views

Probability that one part of a randomly cut equilateral triangle covers the other without flipping

At Probability that one part of a randomly cut equilateral triangle covers the other, the case with flipping allowed was quickly solved. The case without flipping seems more difficult and hasn't been ...
7
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1answer
120 views

Probability that one part of a randomly cut equilateral triangle covers the other

If you make a straight cut through a square, one part can always be made to cover the other. (This is true by symmetry if the cut goes through the centre, and if it doesn't, you can shift it to the ...
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0answers
25 views

Division of Solid Angle When Subdividing Spherical Triangle

Suppose I have a spherical triangle (no special properties; in particular, not equilateral) with a known solid angle. Now, I divide it into four new spherical triangles by bisecting each edge: ...
0
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1answer
47 views

Proving the Pythagorean Theorem with just variables

I basically have just three problems: a) How many similar triangles can you find in the figure below? b) Use part a) to prove the Pythagorean Theorem (note: to prove something you do need to ...
101
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4answers
13k views

What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

My friend gave me this puzzle: What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges? I tried to ...
1
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1answer
37 views

Area of Traingle Problem

In Triangle $DEF$, $P$ is mid point of $EF$ and $Q$ is the midpoint of $DP$. The area of triangle $DQF$ is $6 \ cm^2$. We need to find the area of triangle $EQF$. I tried many ways to solve it but ...
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3answers
38 views

Triangle: Finding $x$ and $y$ (2 sides are given) - 6th grade

I am helping my son with this homework and I was wondering if I can get a tip or few. The question is: Write and solve equations to determine the values of $x$ and $y$. (see picture attached) The ...
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2answers
53 views

value of X and Y from triangle

my son is in 6th grade and i am trying to help him solve this problem. but i want to understand so i can teach him. Write and solve equations to determine the value of x and y . triangle is given (...
0
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1answer
29 views

For a given triangle, prove that $DL=DM$

In a triangle $ABC$, $D$ is midpoint of the side $BC$. Through the point $A$, $PQ$ is any straight line. The perpendiculars from the points $B$, $C$ and $D$ on $PQ$ are $BL$, $CM$ and $DN$ ...