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. anemone

    MHB Finding $\angle APB$ in $PQR$ Triangle with $QA:AB:BR = 3:5:4$

    Let $PQR$ be a triangle with $\angle P=90^{\circ}$ and $PQ=PR$. Let $A$ and $B$ be points on the segment $QR$ such that $QA:AB:BR=3:5:4$. Find $\angle APB$.
  2. N

    Does this spherical triangle exist?

    I am trying to solve the 'ant and honey problem on a spherical bowl' to find the shortest route between two points on a sphere when the path is constrained by not being allowed to pass higher than a certain latitude (so interrupting some great circles connecting the two points). I intuitively...
  3. T

    Probelm with modeling a triangle

    Hi, i am trying to model a triangle in a square cell based on its dimension. However, i have some problem with parts of the code, i think it might be the logic behind my programming or over definition with the code itself. i have attached the code. the problem lies with the second half of...
  4. L

    Vector Multiplication in a Triangle on the Cartesian Plane

    Homework Statement For the vectors in a Triangle, with a = 16, b = 12, and c = 20 what are (a) the magnitude and (b) the direction of A x B (c) the magnitude and (d) the direction of A x C (e) the magnitude and (f) the direction B x C this is Vector Multiplication. Homework Equations...
  5. anemone

    MHB Finding $\angle QCA$ from Altitude $AM$ of Triangle $ABC$

    If $Q$ is a point on the altitude $AM$ of triangle $ABC$, and that $\angle QBA=20^{\circ}$, $\angle QBC=40^{\circ}$ and $\angle QCB=30^{\circ}$, find $\angle QCA$.
  6. T

    Find the length of one of the sides in a triangle

    Hey! I just started study landsurveying and got a task i don't figure out how to get the right answer. My goal is to find the length of one of the sides in a triangle, but i can not use pytagoras formula. Here is the formula i am supposed to use (dont care about the text, its norwegian)...
  7. Dethrone

    MHB Minimizing Isosceles triangle with a circle inscribed

    Find the smallest possible area of an isosceles triangle that has a circle of radius $r$ inside it. I cannot seem to find the relationship between the circle and triangle. Any hints? I'm thinking similar triangles, but I want to know if they're any other approaches before I try that.
  8. S

    Drawing a Triangle with Side a=4 cm, Altitude 3 cm, and Angle α=60°

    Homework Statement Side ##a=4 cm##, altitude to side a is 3 cm , angle ##\alpha =60 °##. How can I draw that? Step by step Homework Equations The Attempt at a Solution
  9. L

    How to find the area of an n-dimensional triangle?

    Homework Statement How to find area of an n-dimensional triangle using vectors? Homework Equations The Attempt at a Solution
  10. anemone

    MHB Equilateral Triangle Intersecting Lines Theorem

    Let $ABC$ be an equilateral triangle, and let $K$ be a point in its interior. Let the line $AK,\,BK,\,CK$ meet the sides of $BC,\,CA,\,AB$ in the points $A',\,B',\,C'$ respectively. Prove that $A'B'\cdot B'C'\cdot C'A' \ge A'B\cdot B'C\cdot C'A$.
  11. S

    Finding the co-ordinates of the 3rd point of a triangle given this?

    On this triangle: Show a formula for finding co-ordinates of B. You know: - the co-ordinates of A and C; - Angle B = 90 degrees Is this possible? If not, is it possible if you know all 3 angles?
  12. anemone

    MHB Triangle Challenge: Prove $p^4+q^4+r^4-2p^2q^2-2q^2r^2-2r^2p^2<0$

    Prove that $p^4+q^4+r^4-2p^2q^2-2q^2r^2-2r^2p^2<0$ for $p,\,q,\,r$ are the sides of a triangle.
  13. A

    Electric field strength at the center of an equilateral triangle

    Homework Statement 3 rods each of length 1 meter form an equilateral triangle. Two rods have a uniform charge distribution of 8\times 10^{-6} C and the third of -8\times 10^{-6} C. What is the electric field strength at the center of an equilateral triangle Homework Equations \vec{E} =...
  14. anemone

    MHB Prove $x^3+y^3+3xyz>z^3$ for Triangle Sides

    Let $x,\,y,\,z$ be the sides of a triangle. Prove that $x^3+y^3+3xyz>z^3$.
  15. M

    Electric Fields -- Charges at corners of equilateral triangle

    Consider an equilateral triangle of side 15.6 cm. A charge of +2.0uc is placed at one vertex and charges of -4.0C uc each are placed at the other two, as shown in the diagram to the right. Determine the electric field at the centre of the triangle ANgle= 60 sides--> d1= d2=d3=0.156m...
  16. M

    Electric Field at Center of Equilateral Triangle

    Consider an equilateral triangle of side 15.6 cm. A charge of +2.0uc is placed at one vertex and charges of -4.0C uc each are placed at the other two, as shown in the diagram to the right. Determine the electric field at the centre of the triangle ANgle= 60 sides--> d1= d2=d3=0.156m...
  17. Greg Bernhardt

    How Does the Vector Triangle Principle Apply to Different Physical Quantities?

    Definition/Summary Vectors (such as velocity or force or momentum) obey the Vector Law of Addition. That means in particular that combination of two vectors \vec{V}_1 and \vec{V}_2 is a vector \vec{V}_3 only if three directed lines (lines with arrows) \vec{L}_{AB} \vec{L}_{BC} and...
  18. A

    Calculating the Centroid of a Triangle: Tips and Tricks

    Hi friends, I have been trying to calculate the centroid of triangle (show in attachment) . I have got centroidal Y= h/3 Not able to get centroidal X = (1/3)*(a+b) Could anyone help on this stuff? Thanks..
  19. S

    Electric Force: Equilateral Triangle

    Homework Statement Three charges, +Q, +Q, and -Q are placed at the vertices of an equilateral triangle of length "s" on a side. Find the magnitude and direction of the force on one of the +Q charges. Homework Equations ForceElectric=(K*q1*q2/d2) The Attempt at a Solution I've set...
  20. S

    Can You Find All Triangle Angles from 2 Sides?

    Homework Statement Is it possible to determine all the angles in a triangle, if we only know the length of two sides?Homework Equations The Attempt at a Solution I was thinking for quite some time and I don't think it is possible. It probably is, if two sides are peprendicular but if not, I...
  21. Lebombo

    Area of a triangle (cross product lesson)

    Homework Statement Youtube: Lec 2 | MIT 18.02 Multivariable Calculus, Fall 2007 (Video time frame: between 11:00 minutes and 12:30 minutes) Find the area of a triangle. Area = \frac{1}{2}(base)(height) = \frac{1}{2}|a||b|sinθ The lecturer says to first find cosine of the angle using dot...
  22. J

    MHB 30-60-90 triangle side lengths

    I have a 30-60-90 triangle with the length of 8 for the long leg. I am trying to find the lengths of the other two legs. I believe the short leg is x, and hypotenuse is 2x, and the long leg is x times the \sqrt{3}. I put x times \sqrt{3}=8 although I am not sure how to do this formula to...
  23. V

    Coefficient of performance triangle process thermodynamics

    Homework Statement An ideal gas adiabatic coefficient γ is submitted to the ABCA cycle of fig, where AB is a line segment.. a) Calculate the income. b) Show that it is smaller than the yield of a Carnot cycle operating between the same temperature extremes. this is my attempt...
  24. Q

    A triangle with a Force applied at one tip

    Homework Statement This is a general question about Statics. I was not able to find a specific question that includes this situation. I have a right triangle ABC with two (or three) members. Member AC is diagonal with a pin support (prevents translation) at C. Member AB is horizontal with...
  25. Albert1

    MHB Max Area of $\triangle ABC$ with $AB=AC$ and $BD=m$

    $\triangle ABC, \,\, AB=AC$,point $D$ is the midpoint of $AC$ if $BD=m$,and $n$=area of $\triangle ABC$ please find $max(n)$ and corresponding $\angle A$
  26. S

    Plus sign in triangle usage (or minus, times)

    Can anyone refer me to a paper where the "plus sign in triangle", "minus sign in triangle", and/or "multiplication sign in triangle" are used? I'm reading a paper from the 80's where these symbols act on an image ("internal law" and "external law"); however, I'd like to see the symbols applied...
  27. O

    Sierpinski Triangle: Equilateral or Isosceles?

    Is the Sierpinski triangle composed of equilateral or isosceles triangles? I've seen references for both but I have a student asking which one it is... any help is here greatly appreciated. Isosceles example...
  28. A

    Coordinate geometry with area of triangle

    Let A(1, 2), B (3,4), C( x, y) be points such that (x- 1) (x-3) +(y-2) (y-4)=0. Area of triangle ABC=1. maximum number of positions of C in the xy plane is (a) 2 (b) 4 (c) 8 (d) None of these I have tried using the staircase formula which gives me something like x-y=2. Therefore I see only...
  29. G

    Finding the Area of a Similar Right Triangle

    For example: if it was given that two right triangles are similar triangles and that the hypotenuse of one is twice as long than the other how would you find the area of the triangle with the twice as long hypotenuse given the area of the other? Similar right triangles means they are the same...
  30. maistral

    Sides of a triangle given an area and an equation.

    WARNING: THIS IS NOT HOMEWORK~! Okay, so the problem goes like this: "Find a,b,c of a triangle; If a+b+c = 10 ; Area = 10" I know it sounds totally vague (I think so too). So I tried using the Pythagorean theorem; c2 = a2+b2 then the given equation; 10 - a - b = c; then the...
  31. anemone

    MHB Triangle Inequality: $a^4-1, a^4+a^3+2a^2+a+1, 2a^3+a^2+2a+1$

    Show that for all $a>1$, there is a triangle with sides $a^4-1$, $a^4+a^3+2a^2+a+1$, and $2a^3+a^2+2a+1$.
  32. M

    Help with Triangle Wave using complex exponential Fourier Series

    I'm participating in research this summer and it's has to do with the Fourier Series. My professor wanted to give me practice problems before I actually started on the research. He gave me a square wave and I solved that one without many problems, but this triangle wave is another story. I've...
  33. adjacent

    Area of Triangle: Find Points A, B, & C

    Homework Statement Find the area of triangle formed by the points ##A(5,2)## , ##B(4,7)## , ##C(7,-4)## Homework Equations Nah The Attempt at a Solution Is there any better way than finding the angle between lines and their lengths and then the area?
  34. M

    Dot Product of Equilateral Triangle

    Homework Statement In an equilateral triangle with sides u, v, w, where they are all unit vectors, find u dot w. Homework Equations u dot w = |u||w|cosθ The Attempt at a Solution The answer is ##\frac {-1} {2} ## cos(120) = -1/2 Elsewhere, I read the statement that since these are...
  35. anemone

    MHB Solve the triangle PQR by finding their angles

    $QB$ and $RA$ are angle bisectors of the triangle $PQR$. Given that $\angle QBA=24^{\circ}$ and $\angle RAD=18^{\circ}$. Find the measure of each angles $P,\,Q$ and $R$.
  36. M

    Proving Aa+ Bb+ Cc = 0 in a Plane Triangle

    Homework Statement Let A, B, C be the vertices of a triangle in the plane and let a, b, c be respectively, the midpoints of the opposite sides. Show that Aa+ Bb+ Cc = 0 (all of them have vector signs on the left). Homework Equations definition of plane The Attempt at a Solution...
  37. S

    Area of a Triangle Using Vectors.

    Homework Statement A triangle has verticies A(-2,1,3), B(7,8,-4), and C(5,0,2). Determine the area of the triange ABC. The correct answer is 35.9 square units. Homework Equations Has to be done by using dot product and/or cross product. Dot product: a(dot) b= |a||b|cos(theta) Cross...
  38. Albert1

    MHB What is the Area of Triangle ODG?

  39. anemone

    MHB How is the Equation Derived for an Isosceles Triangle with a specific Angle?

    Given a triangle $PQR$ where $QR=m$, $PQ=PR=n$ and $\angle P=\dfrac{\pi}{7}$. Show that $m^4-3m^2n^2-mn^3+n^4=0$.
  40. anemone

    MHB Triangle $PQR$: Find $\tan P,\,\tan Q,\,\tan R$ Values

    In triangle $PQR$, $\tan P,\,\tan Q,\,\tan R$ are integers, find their values.
  41. anemone

    MHB Prove $BD=2CD$ in Triangle $ABC$

    In a triangle $ABC$, it's given that $AB=AC$, point $D$ is on $BC$ whereas point $E$ is on $AD$ such that $\angle BED=2 \angle CED=\angle BAC$. Prove that $BD=2CD$. Note: This problem is actually posted by Albert at another math forum about 2 years ago and for all information, I have gained...
  42. S

    Where is the right triangle in sides a, b, c, for a hyperbola?

    An ellipse has some model standard form values, a, b, and c which are easily enough to identify from the graph and parts of the graph related to the ellipse's graph. Seeing the right triangle relating a, b, and c, is easy enough. The Pythagorean Theorem is used to relate these three values...
  43. anemone

    MHB Prove one of the angles of triangle is 60°

    Given that $PA,\,QB,\,RC$ are the altitudes of the acute triangle $PQR$ such that $9\vec{PA}+4\vec{QB}+7\vec{RC}=0$. Show that one of the angles of triangle $PQR$ is $60^{\circ}$.
  44. D

    I'm forgetting my geometry. Can I solve this triangle?

    Homework Statement So I'm doing a physics problem, and I think I'd get the right answer if I solved this triangle. I don't know any angles. The base is 56 meters long, its height is 500 meters, and the difference between the other two sides is 4 meters. Can I figure out the sides based on this...
  45. M

    Area of a right triangle with little data

    Homework Statement Homework Equations x^2 + y^2 = 9 A = 0.5xy x ≠ y The Attempt at a Solution x^2 + y^2 = 9 A = xy/2 (x + y)^2 = x^2 + 2xy + y^2 = 9 + 2xy = 9 + 4A A = ((x+y)^2 - 9)/4 Then I am lost. I need to find the area.
  46. A

    Lorentz triangle and length contraction perpendicular to propagation

    Consider a pole of 1 light second long in the ##y## direction (the vertical line(s) in the enclosed figure). It is moving in the ##-x## direction. According SR, the pole's length is not contracted because its length is not parallel to the propagation direction. However, given the time of flight...
  47. Saitama

    MHB Maximum area of triangle and quadrilateral - given perimeter

    Problem: Let $0<a<b$ i)Show that amongst the triangles with base $a$ and perimeter $a+b$, the maximum area is obtained when the other two sides have equal length $b/2$. ii)Using the result (i) or otherwise show that amongst the quadrilateral of given perimeter, the square has maximum area...
  48. bsmithysmith

    MHB How to find the Double Angle formula for Sin given only a triangle

    I know the Double Angle for Sine is: \sin(2x) = 2\sin(x) \cos(x) but from the triangle given, how do I figure it out? We did this in class, but the teacher just told a small amount of things, and then let us talk amongst each other to solve it. Nearly all the students were glossy eyed and did...
  49. anemone

    MHB Finding Angle P in Isosceles Triangle $PQR$

    Triangle $PQR$ is an isosceles triangle with $PQ=PR$. Given that the angle bisector at $Q$ meets $PR$ at $A$ and that $QR=QA+PA$. Find angle $P$.
  50. Saitama

    MHB Inequality with area of triangle

    Problem: If A is the area and 2s the sum of three sides of a triangle, then: A)$A\leq \frac{s^2}{3\sqrt{3}}$ B)$A=\frac{s^2}{2}$ C)$A>\frac{s^2}{\sqrt{3}}$ D)None Attempt: From heron's formula: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ From AM-GM: $$\frac{s+(s-a)+(s-b)+(s-c)}{4}\geq...
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