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. Calpalned

    Area of a triangle using vectors

    ## 1. Homework Statement Let P = (1,1,1), Q = (0, 3, 1) and R = (0, 1, 4). Find the area of triangle PQR Homework Equations ## \frac {|PQ × PR|}{2} ## = area (The crossproduct divided by two) The Attempt at a Solution I lost my answer key, so I want to check if my final answer of ## \frac...
  2. P

    Prove that a point in triangle is the centroid

    You are given an arbitrary triangle ABC. Inside ABC there is a point M such that Area(ABM) = Area(BCM) = Area(ACM) . Prove that M is the centroid of triangle ABC. I have had very little progress with this question. I've tried connecting a line from M which bisects BC, but I cannot prove that...
  3. anemone

    MHB Triangle Challenge: Prove 2.5<PQ/QR<3

    In a triangle $PQR$ right-angled at $R$, the median through $Q$ bisects the angle between $QP$ and the bisector of $\angle Q$. Prove that $2.5<\dfrac{PQ}{QR}<3$.
  4. T

    MHB Length of a side, possible triangle side

    I have one question from my mathematics for machine technology book that has me stumped as well as my Father in-law, sister in-law and my accountant friend. I wasn't sure if I was supposed to create a right triangle and use A² + B² = C² Problem 18. B...
  5. terryds

    Triangle Ladder equilibrium problem

    Homework Statement http://www.sumoware.com/images/temp/xzebnqpmgdnohmgo.png An isosceles-triangle-shaped ladder like the picture above, has a negligible mass. A man with mass m kg climb the ladder till the height is 3 meter (see picture above). Determine the tension of the link (at...
  6. END

    Electric Field of Point Charges in Equilateral Triangle

    Homework Statement A point charge q =+6.4 \mu \mathrm{C} is placed at each corner of an equilateral triangle with sides 0.29 \mathrm{m} in length. What is the magnitude of the electric field at the midpoint of any of the three sides of the triangle? Hint: if you are careful to examine the...
  7. ChrisVer

    Anomaly cancellation triangle

    If I have a triangle diagram for Z \rightarrow W^-W^+, do all fermions contribute the same for giving the contraint \sum_i Q_i =0?? I think that eg top quarks will be absent for such a diagram... any help?
  8. D

    Maximum area of a triangle inscribed in another triangle?

    Homework Statement [/B] Hello! I have this question which I don't quite know how to solve... ABC is an equilateral triangle - the length of its sides equal to (a). DE is parallel to BC 1. What length should DE be to achieve the largest possible area of triangle BDE? 2. What length should DE...
  9. A

    (Rigorously) Prove that two medians of a triangle intersect

    I'm trying to solve exercise 60 from Kiselev's Planimetry. Homework Statement "Show that in any triangle, every two medians intersect. Is the same true for every two bisectors? altitudes?" Homework Equations There aren't any, unfortunately. The Attempt at a Solution The last part of the...
  10. P

    2 Cubic Blocks on a Right Triangle

    Homework Statement A right-triangular wooden block of mass M is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass m, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger...
  11. anemone

    MHB Lengths of three segments of a triangle

    If $x,\,y,\,z$ are lengths of three segments which can form a triangle, show that the same is true for $\dfrac{1}{x+z},\,\dfrac{1}{y+z},\,\dfrac{1}{x+y}$.
  12. Panda_Doll

    Three Charges in a Triangle problem

    Homework Statement [/B] Three 3.0 g balls are tied to 80-cm-long threads and hung from a single fixed point. Each of the balls is given the same charge q. At equilibrium, the three balls form an equilateral triangle in a horizontal plane with 20 cm sides. Homework Equations F=kq1q2/r^2 F (of...
  13. A

    MHB Given 3 sides of a triangle, compute interior angles and area

    Just attempted another exam question. Would you mind correcting me if I am wrong? The question is A triangle ABC has sides of length AB= 3.5 m, BC = 5.1 m and AC = 4.2m. a) Calculate the size of the angle B and the size of the angle C, in degrees correct to 1 decimal place, in each case...
  14. A

    MHB Why Is Cosine Used to Calculate Angle Y in a Right Triangle?

    Hi, this would be my first post of many in recent times as I have my maths exam soon! I am doing a lot of past papers and I need some help understand some questions. In the triangle XYZ angle Z is a right angle. If XY= 15mm and YZ=8mm, calculate the angle Y, giving your answer in degrees...
  15. agnishom

    Three Springs in an Equilateral Triangle

    Homework Statement [/B] Three identical point masses of mass m each are placed at the vertices of an equilateral triangle and joined through springs of equal length and spring constant k . The system is placed on a smooth table. If the masses are displaced a little towards the centroid of the...
  16. ltkach2015

    Alternative Vector Method to Finding Angles of Vertices of Triangle?

    NOTE: I know the solution, but I challenge it. Specifically, I feel that the generally accepted solution to this problem is not intuitive. I have shown the accepted solution, and below that I have shown what I feel seems more intuitive (from a geometric standpoint)...
  17. C

    Proving the Sum of Cosines in a Triangle Using Half-Angle Formula

    Prove for any triangle ABC that: cos(A)+cos(B)+cos(C)=1+4sin(\frac{1}{2}A)sin(\frac{1}{2}B)sin(\frac{1}{2}C) I tried using the half-angle formula on the right side to get: 1+4 \left(\frac{1-cos(A)}{2}\frac{1-cos(b)}{2}sin(\frac{1}{2}C)\right) {} which simplifies to...
  18. C

    Center of gravity of a 3d triangle using weight

    Hi all, I am after some help to find out how to do the following: I have a triangular object (weighing around 40,000kg) which I can measure and weigh. I have 4 load indicators I can position anywhere beneath the object, these will give me a very accurate reading of the weight of the object at...
  19. anemone

    MHB What is the area of triangle PQR divided into six smaller triangles?

    As shown in the figure below, triangle $PQR$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles area as indicated. Find the area of triangle $PQR$.
  20. H

    [Double integral] Area of a triangle

    Hi! I'm stuck with the following problem: ----------------------------------- Calculate ∫∫ (x-y)*|ln(x+2y)| dxdy where D is the triangle with corners in the coordinates (0,0), (1,1) and (-3,3) ----------------------------------- I get the following lines that forms the triangle: y=-x, y=x...
  21. H

    Total electric potential energy of equilateral triangle

    Three charged objects, (+4 micro coulomb, -4 micro coulomb and +2 micro coulomb) are placed at the corners of an equilateral triangle with side length 2m. Calculate the total electric potential energy of the system... Guys Can you help me achieve a solution to this question? Thanks In Advance!
  22. CivilSigma

    Calculating Centroids of a Triangle: Why Do Formulas Switch?

    Homework Statement Hello. In class we learned how to calculate centroids of a triangle. For the y-coordinate we use the formula height/3 and for the x we use 2/3 * base Homework Equations In the diagram, If I want to calculate the y coordinate , I do 72 (which is the proposed height)/ 3 ; only...
  23. G

    Center of mass of an equilateral triangle (Kleppner)

    Homework Statement Find the center of mass of an equilateral triangle with side ##a## Homework Equations ## \vec R = \frac{1}{M} \int \vec r \ dm ## ## dm = \frac{M}{A} dx dy ## ## A = \frac{\sqrt{3}}{4}a^2 ## The Attempt at a Solution I set a pair of orthogonal axis ##(\vec x,\vec y)##...
  24. dawo0

    Equilateral triangle ABC is inscribed in a circle

    Homework Statement Can someone help me solve this, and teach me how to solve such problems in future? An equilateral triangle ABC is inscribed in a circle . Point D lies on a shorter arc of a circle BC. Point E is symmetrical the point B relating to the line CD . Prove that the points A, D , E...
  25. I

    MHB Integrating y^3 over Triangle D

    $\int \, \int_{D}^{} \, y^3 dA$ D is the triangular region with vertices (0,1), (1,2), (4,1) i can't get past this problem. i drew the triangle but i don't know how to find the intervals... excuse my ugly drawing :p
  26. EverGreen1231

    Questions pertaining to Transformer DGA and Duval Triangle

    Is anyone versed enough in Power transformer DGA and Duval triangles to know what the implications of the different regions might be? I've Googled and found many documents about the subject, but none really seem to mention what the implications might be for a transformer that tends towards T3 as...
  27. anemone

    MHB Prove a triangle is equilateral

    Let $a,\,b,\,c$ be the sides and $A,\,B,\,C$ be the opposite angles angles of a triangle. Show that if $ab^2\cos A=bc^2\cos B=ca^2\cos C$ then the triangle is equilateral.
  28. E

    MHB Distance Between Intersection Points of Bisectors & Medians in Right Triangle

    The legs of chateti of a right triangle are 9 and 12 cm. Find the distance between the intersection point of bisectors and the point of intersection of the medians
  29. S

    Task1: Vector Triangle: Calculate Resulting Speed

    I'm having my semester test tomorrow, and I got this task from my teacher to do before the semester test. : By decomposition of forces apply these formulas for a right triangle: Task1: An iceberg floats with the stream and wind. The power comes straight from the north with speed 3m / s. Draw...
  30. JasonHathaway

    The area of a square with right triangle inside it

    Homework Statement I need to find the area of the square in the following figure: Homework Equations Basic Trig relations. The Attempt at a Solution I aimed to find the length of BC, but first I had to find the unknowns of the right triangle CDE, which are EC=5m, <DCE=36.86ْ , <DEC=53.13ْ ...
  31. I

    MHB Modify Triangle Program #2 - Enter Valid Sides for Area

    ok so in this program, if three valid sides are given it displays the area of a triangle otherwise it will ask the user to re enter sides. #include <iostream> #include <cmath> using namespace std; int main(){ double a,b,c=0; double s,A=0; cin>>a>>b>>c; cout<<"Enter three sides: "<<a<<"...
  32. G

    MHB Right Angle Triangle, at a loss

    So I have calculated Sin(70°) = Opp / 30.816, in this equation Opp = 28.95757 BUT when I do Sin(X) = 28.95757 / 30.816 I result in 1.08422336464 when converted to degrees gives me 62.12° NOT 70° If I put Opp (a) (28.95757) and Hypotenuse of (30.816) into the following calculator it says 70°...
  33. T

    MHB How Fast Does the Area of a Triangle Change with Increasing Angle?

    Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $$\pi/3$$ When the angle is $$pi/3$$, the third side is equal to...
  34. I

    C/C++ Compute the area of a triangle c++

    compute the area of a triangle. write an if/else statement to check if the three sides provide a valid triangle. to make a valid triangle, all sides have to be positive and the sum of any two sides must be greater than the third. add a while loop so that if the sides are invalid the user is...
  35. paulmdrdo1

    MHB Area of Isosceles Triangle proof

    I want to know how did it arrive at the third line of this proof. I didn't get it. Thanks!
  36. paulmdrdo1

    MHB Proving Inscribed Angle Theorem for Triangle Side Lengths

    I already found the side lengths that's being asked in the problem. But the part that I'm having difficulty is the Proving part. Can you help me do that? Thanks!
  37. paulmdrdo1

    MHB Finding the lengths and altitudes of s triangle

    For a certain triangle ABC, sin A = 12/13, tan B = 15/8, and the altitude to side AB is 60 units. Find the lengths of the sides and of the altitudes of the triangle. Again, I couldn't come up with an image of the triangle in question. Maybe you can provide me the picture of the problem so that...
  38. R

    Center E field of an equilateral triangle.

    y/SUB]Homework Statement Determine the electric field at the center of the triangle. All sides are 15.6 cm. Vetex is 2.0 uC (Q1) while the bottom points are both -4.0 uC (Q2 and Q3). Homework Equations E = kQ/r^2 E = F/q The Attempt at a Solution Ecenter = E1 + (2)E2 = ? First calculate...
  39. C

    Maximum and Minimum Values Inside Triangle

    Homework Statement Find the absolute maximum and minimum values of f(x,y) = y^2+x^2 -4x + 11 on the set D where D is the closed triangular region with vertices (8,0),(0,4) , and (0,-4) . Homework EquationsThe Attempt at a Solution [/B] fx = 2x - 4 fy = 2y Critical point = (2,0) The...
  40. M

    Solving gradients of right angle triangle

    Please take a look at attached diagram displaying gradients of each line, please explain how m3 is obtained.
  41. R

    Optimization of the Area of a Triangle

    Homework Statement Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 4. Solve by writing the area as a function of θ Homework Equations A=1/2 (bh) The Attempt at a Solution Given the side h and the hypotenuse 4, we can find the base of the smallest...
  42. R

    Kinematics Problem -- 3 particles at the vertices of an equilateral triangle

    Here's the question: Three particles A, B and C are situated at the vertices of an equilateral triangle ABC of side d at time t = 0. Each of the particles moves with constant speed v. The particle at A always has its velocity along AB, B along BC and C along CA. At what time will the particles...
  43. anemone

    MHB Determine the type of triangle PQR

    Determine all triangles $PQR$ for which $\cos P \cos Q+\sin P \sin Q \sin R=1$.
  44. T

    How to rotate a triangle in an equation

    Hi, i have a set of definition for a triangle shape: -3/4<x<3/4 -x/sqrt(3) + sqrt(3)/4 < y < x/sqrt(3) +sqrt(3)/4 The centroid of this triangle is at 0,0 i want to rotate this triangle by a certain degree from the centroid but i am not sure how, because if i just multiply it by cos or sin...
  45. karush

    MHB *find eq of perpendicular line so that the area of the triangle is 8

    for the line y=4x-2 there is one perpendicular line of which will enclose a triangle on the lines and the values of the y-axis whose area is 8. What is the equation of this line? Well, I chose the x value to be the height of the triangle and that would make the base B=(4x-2)+2+\frac{1}{4}x or...
  46. M

    Electric Flux of equilateral triangle

    Homework Statement What is the flux of the uniform electric field E=8 N/C through an equilateral triangle with the side a=1m, if the normal vector makes 60 degrees to the direction of the electric field. Homework Equations \phi=EAcosøThe Attempt at a Solution E=8 N/C A= √3/4 m^2 Is the ø in...
  47. J

    Mechanical force question on equilateral triangle

    Homework Statement Three particles with masses m1=2.5kg, m2=1.2kg and unknown mass m3 are placed at the corners of an equilateral triangle of side 30.0cm. If the resultant force in x-direction acting on m1 is zero, find the net force acting on m1. Homework Equations F=(Gm1m2)/R^2...
  48. A

    Find the flux through the equilateral triangle with corners at

    Homework Statement Find the flux through the equilateral triangle with corners at the points (1m,0,0), (0,1m,0), and (0,0,1m) in x,y,z space (measured in meters) for an electric field with magnitude E=6N/C pointing (a) in the z direction, (b) parallel to the line y = x. Homework...
  49. J

    The triangle inequality in CHSH, where is the triangle?

    http://en.wikipedia.org/wiki/CHSH_inequality#Bell.27s_1971_derivation The last step of the CHSH inequality derivation is to apply the triangle inequality. I see there are relative polarization angles, but I don't see any sides have defined length to make up a triangle. Where is the triangle?
  50. P

    Finding the Nonzero Vector and Area of Triangle

    Homework Statement Find a nonzero vector orthogonal to the plane through the point P, Q,and R. (b) also find the area of triangle PQR P(1,0,1) , Q(-2,1,3) , R(4,2,5) Homework Equations -Cross product -Finding the Angle -Area formula The Attempt at a Solution My steps: 1. i found the...
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