What is Triangle: Definition and 1000 Discussions

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted



△
A
B
C


{\displaystyle \triangle ABC}
.In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted.

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  1. modularmonads

    [Euclidean Geometry] Kiselev's Plainimetry Question 242

    Homework Statement Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points D and Ε with MA and MB respectively. Prove that (1) the perimeter of ▲DME, and (2)...
  2. anemone

    MHB Right Triangle: Proving It's a Right Triangle

    The sides $x,\,y,\,z$ of a triangle satisfy the equality $(x^4+y^4+z^4)^2=2(x^3+y^3+z^3)$. Prove that it's a right triangle.
  3. R

    Proving Triangle Side Lengths Using Congruence and Similarity

    Homework Statement As shown in the diagram below, the shape consists of a square and a circle with centre Q. Given that QM = 3 cm, prove that MN = 6 cm. Known data: -- triangles APB and BQC are congruent -- angle BMC = 90 -- triangles BMQ and BNA are similar and right-angled 2. Homework...
  4. terryds

    Given isosceles triangle, find sin (A-C)

    Homework Statement Triangle ABC have side AB = 10 cm, AC=BC = 13cm, so sin (A-C) is... 2. Homework Equations sin (A-C) = sin A cos C - cos A sin C The Attempt at a Solution I see that the triangle can be split into two right-angle triangles. But, sin (A-C) ?? How to get that?[/B]
  5. O

    MHB Generalized triangle inequality in b-metric spaces

    How is the generalized triangle inequality in b-metric spaces ? I find something...But I wonder your opinion...Thank you for your attention... Especially if you write for n,m>0 m>n $d({x}_{n},{x}_{m})$$\le$..... I will be happy...
  6. O

    MHB Triangle inequality in b-metric spaces

    Let $X$ be a non-empty set and let $s\ge1$ be a given real number. A function $d:$ X $\times$ X$\to$ ${R}^{+}$ , is called a b-metric provided that, for all x,y,z $\in$ X, 1) d(x,y)=0 iff x=y, 2)d(x,y)=d(y,x), 3)d(x,z)$\le$s[d(x,y)+d(y,z)]. A pair (X,d) is called b-metric space. İt is clear...
  7. Y

    Area of Triangle ABC: Find Solution

    Homework Statement In triangle ABC, ∠C=90∘. Let D, E, F be points on sides BC, AC, AB, respectively, so that quadrilateral CDFE is a rectangle. If [BDF]=7 and [AEF]=28, then find [ABC]. Homework Equations Area of a rectangle, and triangle. Also can cut up the rectangle into some triangles...
  8. W

    MHB Triangle Questions: Get Expert Advice Now

    Hi everyone I need help in these 2 questions. Please advice. Tis :)
  9. enter

    Getting a triangle from trigonometric function

    How can I get a right triangle from the inputs and outputs of trigonometric functions? For example: sin(x) = y The triangle would have one angle as x and the opposite edge of the triangle would be y/hyp etc. How can I get all of these values from any trigonometric function? Please tell me if I...
  10. Y

    {Geometry} Find length of the equilateral triangle

    Homework Statement http://i.imgur.com/lnk7e0D.png CDE is an equilateral triangle inside a circle, with side length 16. FGH is also an equilateral triangle and F is the mid point of DE. Find the length of the side FGH. Should be expressed as Asqrt(B)-C. Where A B and C are positive integers...
  11. ognik

    MHB Reverse triangle inequality with a + sign

    Thought I knew this, but am confused by the following example: Show $ |z^3 - 5iz + 4| \ge 8 $ The example goes on: $ |z^3 - 5iz + 4| \ge ||z^3 - 5iz| - |4|| $, using the reverse triangle inequality It's probably right, but I don't get why the +4 can just be made into a -4 ?
  12. Rectifier

    Calculate the Unknown Angle of a Right Triangle

    The problem A right triangle has an angle a and we know that ##cos \ a = \frac{1}{3}##. What is ## tan \ (90°-a) ## The attempt I know that the ration between the adjacent side and the hypothenuse is 1/3. I am not interested in the real lengths of the sides. I can therefore calculate the...
  13. H

    Electrostatics: 3 Charges In An Equilateral Triangle

    Hello, so this was a past assignment question that I attempted. I got the right answer but in an incorrect way (I guess my thinking was not based on the concept?) 1. Homework Statement Three point charges of charge Q = 10...
  14. Y

    Maximize the area of a triangle

    http://imgur.com/Q5gjaSG Consider the semicircle with radius 1, the diameter is AB. Let C be a point on the semicircle and D the projection of C onto AB. Maximize the area of the triangle BDC. What I'm thinking y=sqrt(r^2-x^2) From the formula of a circle x^2+y^2=r^2 A=1/2(x+1)y The area of...
  15. M

    MHB Finding the measure of Triangle ABC

    A = 46 degrees b = 8 I don't even know how to start this and I'm really confused. I've already labeled A and b but I really have no clue on how to continue. .-. Can someone please explain this very carefully to me and use simple terms? I've been trying to do this problem and I was told that I...
  16. A

    MHB How Do You Solve a Triangle Problem on a Number Plane?

    Please just hint me in the right direction, I'm kind of lost with it. Thanks for any help
  17. C

    Angles between sides of triangle ABC and unit vectors

    I was going through this link -...
  18. C

    Resultant of 3 vectors along the sides of an equilateral triangle

    Homework Statement Hi all, It is a homework problem, but I really don't quite understand the question. It reads- "3 forces of magnitudes 10N, 20N, and 30N acting on a point are parallel to the sides of an equilateral triangle, taken in order. Find their resultant"Homework EquationsThe Attempt...
  19. rpthomps

    Rotational Inertia of a triangle

    Homework Statement A thin, uniform vane of mass M is in the shape of a right triangle, as shown. Find the rotational inertia about a vertical axis through its apex, as shown in the figure. Express your answer in terms of the triangle’s base width b and its mass M. Homework EquationsThe...
  20. Albert1

    MHB Find the area of the triangle

    $a,b,c,d >0$, please prove we can construct an triangle with length: $\sqrt{b^2+c^2},\sqrt{a^2+c^2+d^2+2ac},\sqrt{a^2+b^2+d^2+2bd}$ and find the area of the triangle
  21. noowutah

    Isosceles triangle in information theory

    In Euclidean geometry (presumably also in non-Euclidean geometry), the part of the dissecting line that dissects the vertex angle and is inside the isosceles triangle is shorter than the legs of the isosceles triangle. Let ABC be an isosceles triangle with AB being the base. Then, for...
  22. S

    Double integral on triangle using polar coordinates

    Homework Statement Let R be the triangle defined by -xtanα≤y≤xtanα and x≤1 where α is an acute angle sketch the triangle and calculate ∫∫R (x2+y2)dA using polar coordinates hint: the substitution u=tanθ may help you evaluate the integral Homework EquationsThe Attempt at a Solution so the...
  23. L

    MHB Very dificult: The minimum perimeter and maximum height of a triangle under constraints

    Obtain -The maximum height corresponding to the side b of any triangle (abc) once known the value of its perimeter and height corresponding to the a side a. -The minimum perimeter of any triangle (abc) once known the heights corresponding to the a and b sides. Aux: Geogebra construction...
  24. Saracen Rue

    Can an isosceles triangle have 3 equal angles?

    Okay, I know I must sound like a complete idiot here, but please bear with me. I've come across a scenario in which I have triangle ECF. Angle ECF = 60 degrees, Angles CEF and CFE are unknown, lengths EC and FC are unknown and equal and length EF is r√3 and not equal to lengths EC and FC I...
  25. noowutah

    dissecting an isosceles triangle

    Simple question, but I can't figure it out. Consider an isosceles triangle ABC with \alpha=\beta dissected by a line through C and D, where D is on AB. It is obvious that |CD|<=|AC|=|BC|, but I want to prove it using trigonometry. I can use |BD|<=|BC| in my assumptions but not...
  26. R

    The orthocentre of the triangle and a parabola

    Homework Statement The orthocentre of the triangle formed by points t1,t2, t3 on the parabola y2 = 4ax is vertex Origin Focus (1,0) Homework Equations NA The Attempt at a Solution The points can be taken anywhere, So orthocentre can be formed anywhere isn't it?
  27. C

    Path of an object at the vertex of an equilateral triangle

    Homework Statement There are three objects at the Vertices of an equilateral triangle that start movin towards each other at the same time with a speed v. Describe the path of the objects and the time taken for them to meet. Homework Equations V1=v3 - v2 Where all velocities are in...
  28. SonOfGod

    Calculate Triangle Area with Basic Ratio Method

    This is supposed to be a simple question. However, I forgot a lot of the basics and rules I have to follow. I tried to workout the height based on the area: 0.5 x 3 x h = 30 h = 20 But couldn't figure out the rest. Then I thought about going by ratio (not from knowledge but out of...
  29. K

    Center of mass of an inclined triangle

    Homework Statement Where's the COM Homework Equations The COM of a right triangle is a third of an edge apart of the right angle vertex The Attempt at a Solution Edge AC: ##\frac{50}{\cos 20^0}=53.2## Two thirds of edge AB: ##\frac{53.2\cdot 30^0\cdot 2}{3}=30.7## One third of edge BC...
  30. G

    What is the equation for predicting the middle number in Pascal's triangle?

    I am trying to find the equation to predict the next middle number in pascal's triangle. By middle number I mean in each row that has odd number of numbers the middle number of that row. So for example row 6 which has 1,6,15,20( middle number), 15,6,1. I am trying to find that middle number, but...
  31. kaliprasad

    MHB Right Angled Triangle: Find $\sin(\theta)$

    Let $a,b,c$ be the sides of a right angled triangle. Let $\theta$ be the smallest angle of this triangle. If $\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c}$ are also the sides of a right angled triangle then show that $\sin(\theta) = \dfrac{\sqrt{5} - 1}{2}$
  32. L

    Triangle problem, finding unknown sides

    Homework Statement Alexander has a 6.0m long pole. He wants to use the pole to make a right triangle. One of the legs, meaning not the hypotenuse, is 2.0m long. Calculate the length of the tho other sides in the triangle. Homework Equations Phytagorah theorem asquared + b squared is equal to c...
  33. B

    Finding the centroid of a triangle using complex numbers

    Hi all, I'm preparing for a deferred exam this semester after falling ill last year. Just looking over my course notes and have a question. I understand how this works in the big picture scheme. What I don't understand however is how my instructor simplified the original equation. 1. Homework...
  34. M

    Side lengths of inscribed triangle

    Homework Statement The corners A, B and C of a triangle lies on a circle with radius 3. We say the triangle is inscribed in the circle. ∠A is 40° and ∠B is 80°. Find the length of the sides AB, BC and AC. Homework EquationsThe Attempt at a Solution I found out the arc AB is 2π, arc BC is 4π/3...
  35. L

    Geometry problem, area of a triangle

    Homework Statement One of the sides of a triangle is 7.0cm, another side is 11.0cm. A Decide the biggest area this triangle can have. B Make calculations and show how the triangle could look like if the area is 30 square cm. Homework Equations Area of a triangle: 0.5*g*h or 0.5*a*b*sinV The...
  36. T

    Solving Triangles. My answer fluctuates from the real answer

    Hey Guys. I'm having a bit of a problem with my solving triangles book. I'm finding the book really easy but there's this one thing that I keep getting wrong. Whenever I'm working with degrees with decimal points my answer aways fluctuates slightly from the real answer. I must be doing something...
  37. B

    RMS and average current triangle

    I would like to know how to find RMS and average current of a triangle? i searched the web. but it focus from 0. which i know. but my waveform is different from the internet. i have attached the waveform.. Maximum = 2.5 , Minimum = 1.8 i need to find out the RMS and average current
  38. T

    Three electrons form an equilateral triangle

    1. Homework Statement Three electrons form an equilateral triangle 1.00nm on each side. A proton is at the center of the triangle. What is the potential energy of this group of charges? Known Variables: s = 1.00 × 10-9m p+ Charge = 1.60 × 10-19C e- Charge = -1.60 × 10-19C r = s/√(3) = 5.77 ×...
  39. G

    Magnetic Force of a Right Triangle influenced by a Line

    Homework Statement http://www.sumoware.com/images/temp/xzlknterambqmokp.png How to calculate the Magnetic Force of THE "Right Triangle" influenced by a Line when the magnetic field isn't constant B=u0*I/2piR? Homework Equations B=u0*I/2piR F=iLB The Attempt at a Solution I can use F=iLB to...
  40. T

    How to tell if a triangle has two solutions?

    I've never been quite sure? Is it just a case of trial and error? Or just knowing the limits of sin, cos and tan?
  41. Albert1

    MHB Proving Orthocenter Property of Triangle ABC

    Point $H$ is the orthocenter of $\triangle ABC$ prove :$HA^2+BC^2=HB^2+AC^2=HC^2+AB^2$
  42. R

    Triangle wave Fourier transform

    Hello, Im not sure if it is the right place to ask it but anyway ... i got this function: \begin{equation} M(t)=\sum\limits_{q=1}^N \frac{v^2}{N+ \frac{1}{2}} \cot^2 \left(\frac{\alpha_q}{2}\right) {\sin^2\left(\sin\left(\frac{\alpha_q}{2}\right)t\right)} \end{equation} where: \begin{equation}...
  43. M

    Introduction to Electrostatics -- Positive charges at the corners of a triangle

    Homework Statement Three positive particles of charges 11 μC are located at the corners of an equilateral triangle of side 15.0 cm. Calculate the magnitude and direction of the net force on each particle. Homework Equations Coulomb's Law The Attempt at a Solution I think, that this is easy...
  44. I

    Solving Laplace's Equation for 2D isosceles right triangle

    Homework Statement Find the two-dimensional solution to Laplace's equation inside an isosceles right triangle. The boundary conditions are as is shown in the picture: The length of the bottom and left side of the triangle are both L. Homework Equations Vxx+Vyy=0 V=X(x)Y(y) From the image...
  45. U

    2-form oriented triangle, Differential Forms

    Homework Statement Find the value of the 2-form dxdy+3dxdz on the oriented triangle with (0,0,0) (1,2,3) (1,4,0) in that order. Homework EquationsThe Attempt at a Solution I have tried various subtraction of these coordinates and applying them to the formula but the answer is in the back of...
  46. P

    Three charges at corners of equilateral triangle

    Homework Statement Three point charges q,q and -q lie at three corners of an equilateral triangle of side length d with -q at the apex. If an electron is released from rest midway at the base (point P) what is its KE when its reasonably far away? Homework Equations V = kq/r, U = qV The...
  47. S

    MHB Barycentric coordinates in a triangle - proof

    I want to prove that the barycentric coordinates of a point $P$ inside the triangle with vertices in $(1,0,0), (0,1,0), (0,0,1)$ are distances from $P$ to the sides of the triangle. Let's denote the triangle by $ABC, \ A = (1,0,0), B=(0,1,0), C= (0,0,1)$. We consider triangles $ABP, \ BCP, \...
  48. M

    MHB Using vector notation describe a triangle

    Hey! :o Using vector notation describe a triangle ( in space ) that has as vertices the origin and the endpoints of the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Could you tell me what I am supposed to do?? (Wondering)
  49. Lena Carling

    Net Forces of 3 uneqaul charges of equilateral triangle

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > Three point charges are located at the corners of an equilateral triangle as Q1 in the top, charge Q2 in the left corner and charge Q3 in teh right corner. Each leg of the triangle has a length...
  50. M

    MHB Find Area of Triangle with Vertices $(0, 0, 0), (1, 1, 1)$ and $(0, -2, 3)$

    Hello! :o We have a triangle with vertices $(0, 0, 0), (1, 1, 1)$ and $(0, -2, 3)$. We want to find the area. How could we find it?? Do we maybe use the fact that the area of the triangle is the half of the area of the parallelogram?? (Wondering) How do we know that it stands?? How can we...
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