Triangle Definition and 1000 Threads

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. D

    I Help with area formula using direction cosines

    I am watching a youtube video and the guy is finding the area of triangles in the coordinate planes as shown in the image still below. So the triangle ABC has area a and normal n with direction cosines {L,M,N}. He then says it is obvious from geometry that the area of triangle OAC is given as...
  2. chwala

    Prove the identity of a triangle in the given problem

    My approach, ##\cos^2 \frac{1}{2} A = \dfrac{s(s-a)}{bc}## ... then it follows that, ##\dfrac{1}{a} \cos^2 \dfrac{1}{2} A+ \dfrac{1}{b} \cos^2 \dfrac{1}{2} B +\dfrac{1}{c} \cos^2 \dfrac{1}{2} C =\dfrac{s(s-a)+s(s-b)+s(s-c)}{abc}## ##=\dfrac {3s^2-s(a+b+c)}{abc}## ##=\dfrac{3s^2-2s^2}{abc}##...
  3. F

    I Can you solve unknown triangle from shared hypotenuse?

    My question comes from an accelerometer attached to a rotating plate. C and T are the centrifugal and tangentinal accelerations. X and Y are the accelerometer x, y axis readings. The x and y axis are not perfectly aligned with the C,T accelerations and are rotated by some arbitrairy unknown...
  4. tellmesomething

    Find the limit of the sum of the areas of all these triangles

    Firstly im not sure if the new triangle that we make of height 3 times the original height of an equilateral triangle of side a, will be an equilateral triangle as well or not. I assumed it would be, please let me know if I interpreted the question wrongly. Continuing this train of thought I...
  5. S

    Electric field of point along the central vertical axis of a triangle

    here is my attempted solution. ## d^2 = z^2 + \frac {L^2} {3} ## ## C ## is coulomb constant since the point is symmetric, only the vertical component of the electric field remains. So, $$ E = 3 E_y =3 \frac {C Q cos \theta} {d^2} $$ $$ E= 3 \frac {C Q z} {d^3} $$ thus part (a) is done ( i...
  6. chwala

    Find ##\tan \theta ## in the form of ##a+b\sqrt {2}##

    There is an error on ms ...unless i am missing something.
  7. Hill

    Three points on the Complex plane

    ##arg((b-a)/(c-a))## is an angle between ##ab## and ##ac##. ##arg((a-c)/(b-c))## is an angle between ##ca## and ##cb##. For them to be equal, ##b## has to be equidistant from ##a## and ##c##, i.e. ##|b-a|=|b-c|##. Then the equation for distances becomes, ##|b-a|/|c-a|=|c-a|/|b-a|##. Thus...
  8. walkeraj

    B A Question Relating Sum of Angles and Breaking of the Parallel Postulate

    Question: What is the relationship between the sum of the angles of a non-euclidean triangle being greater or less than 180 degrees and the definite breaking of the parallel postulate? Is the proof of this trivial? Edit: Additionally, can we say that if the angles of a triangle sum to greater...
  9. I

    1st year high school geometry proof

    I have proved that triangles around the equilateral triangle are congruent, but I don't know how to prove that they are arranged in such a way that they actually do form an equilateral triangle. Like, do I write that they do form an equilateral because it's given in the problem that triangles...
  10. C

    Discover Circle Theorem Help for Triangle PST: PSR Angle = 77 Degrees

    From triangle PST, angle PSR = 180-(81+22) =77 degrees sum of angles in triangle. Is my attempt correct?
  11. wrobel

    B A triangle problem: Prove that |AC|+|AB|=|PR|+|PQ| given some constraints

    I actually do not understand where to place this thread. Hope that it is a high school level problem. There are two triangles ABC and PQR. The vertex A is a middle of the side QR. The vertex P is a middle of the side BC. The line QR is a bisector of the angle BAC. The line BC is a bisector of...
  12. YouAreAwesome

    Solve 2 Triangle Problem: Calculate Exact Value of x

    TL;DR Summary: My student had this in an exam. Have I solved it correctly? Question: Calculate the exact value of x. First note we've not been provided any numerical lengths, therefore the expression for x will include at least two variables, one to account for the variable size of ΔCDB, and...
  13. J

    I Prove that a triangle with lattice points cannot be equilateral

    I assumed three points for a triangle P1 = (a, c), P2 = (c, d), P3 = (b, e) and of course: a, b, c, d, e∈Z Using the distance formula between each of the points and setting them equal: \sqrt { (b - a)^2 + (e - d)^2 } = \sqrt { (c - a)^2 + (d - d)^2 } = \sqrt { (b - c)^2 + (e - d)^2 }(e+d)2 =...
  14. H

    Electric field at the center of the equilateral triangle

    I've found the distance from each point to the center, which is equal to r=20x1.732/3 = 11.55 cm. I find out that E2 and E3 due to -4µEyC on x-direction canceled each other. The E2y = E3Y = EY = E2Ycos60 = E2/2 = [(KQ2)/r^2]/2 So the net E-field: E = E1 +E2y+E3Y =kQ1/r^2 + [(KQ2)/r^2]/2 +...
  15. C

    Why is this result different? (calculating the sides of a triangle)

    so basically, here is a photo from the textbook(in attachments) and I'll write here how I did it. In my opinion, results should have been the same, but for some reason, they differ. So, if anyone can tell me what I am doing wrong I would appreciate it since I can't find mistakes caused by wrong...
  16. brotherbobby

    A formula involving the sum of cosines of the angles of a triangle

    Problem Statement : The statement appeared on a website where a different problem was being solved. I got stuck at the (first) statement in the solution that I posted above 👆. Here I copy and paste that statement from the website, which I cannot show : Attempt : To save time typing, I write...
  17. C

    Justifying Equilateral Triangle AEF in ABCD

    Given a rectangle ABCD and points E, F such that the triangles BEC and CF D are equilateral and each of them has only one side in common with the rectangle ABCD. Justify that triangle AEF is also equilateral.
  18. M

    I Exploring the Uses of Triangle Inequality

    Hi, Recently I studied triangle inequality and the proof using textbook precalculus by David Cohen. My question is whats the benefit of this inequality ? One benefit I found is to solve inequality of the form |x+a| + |x+b| < c which make the solution much easier than taking cases. I assume this...
  19. A

    Geometry problem with arcs in a triangle

    In the figure, the point S is located inside the section FE. Starting from S, as indicated in the figure, six circular arcs are drawn step by step around arcs around A, C, B, A, C, B are drawn. Show that the sixth arc leads back to S and that the six arcs together are then exactly as long as the...
  20. Einstein44

    Triangle formed by straight lines

    I tried doing this problem, but couldn't find a solution that wouldn't involve a sketch... Finding the vertices was all I've got. Thanks in advance.
  21. VladStoyanoff

    I Given a triangle ABC, on which side of the triangle does the point D lie?

    So I've got the following problem: I have points A, B, and C which form a triangle in a 3D space (each point of the triangle has x,y, and z coordinates). I need to find out on which side of the triangle point D lies. I do not have access to the normal of the triangle. How am I supposed to...
  22. A

    I Finding the center of area (centroid) of a right triangle

    To find the y value of the centroid of a right triangle we do $$\frac{\int_{0}^{h} ydA}{\int dA} = \frac{\int_{0}^{h} yxdy}{\int dA}$$ What is wrong with using $$\int_{0}^{h} ydA = \int_{0}^{b} y*ydx$$ as the numerator value instead especially since ydx and xdy are equal and where h is height of...
  23. D

    Prove that the altitudes of a triangle are concurrent

    Hi everyone I have the solutions to this problem, but I'm not sure I fully understand them. Is the idea behind the proof that all of the following can only be true if the altitudes meet at O? 1. c-b is perpendicular to a 2. c-a is perpendicular to b 3. b-a is perpendicular to c. That is...
  24. S

    Is this question doable? Finding triangle side length

    Given a triangle ABC, whose area is 5.5 cm square, and length of AB = 2√3 + 1 cm, find the value of length AC. Given triangle has no special properties like isoceles etc. Only 1 length and area of the triangle are given. Is it possible to solve such a question? thanks.
  25. M

    MHB Difference of angles in a triangle

    In a triangle ABC, let D and E be the intersections of the bisectors of the angles ABC and ACB with the sides AC and AB, respectively. Knowing that the measures in degrees of the angles BDE and CED are equal to 24 and 18, respectively, calculate the difference in degrees between the measures of...
  26. M

    MHB Find the angle in the triangle

    I've tried: BC || EF How to find angle GDC? I think GDC=7x but why? I have an answer but how to solve this?
  27. D

    MHB How to Find the Length of the Hypotenuse of a Right Triangle?

    AB=3, DC=5, ∠ CAD=$45^o$, AB ⊥ BC. Find the length of AC.
  28. M

    MHB Proving $\angle ABC$ is Acute: Inside the Triangle $ABC$

    Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.
  29. L

    Triangle Math Help: Calculate Angles & Lengths with 46 & 35 Measurements

    I couldn't find a triangle calculator that input triangle height. Just angles and outer length. I have a triangle with 46 at one length and a line from center of length going to opposite corner that is 35. How can I calculate all angles from that? Sorry. I dumb.
  30. Eobardrush

    Finding area of a non right angled triangle

    I just simply used the formula to solve. Note the "x" represents multiplication in this case 0.5 x a x c Sin B This is based on the conditions given in the textbook I am using which quotes "Use this formula to find the area of any triangle when you know 2 sides and an angle between them" So I...
  31. vibha_ganji

    Three Points at Vertices of Equilateral Triangle

    I’m not sure of how to begin solving this problem. I attempted to draw a diagram and finding the components velocity of each initial velocity vector but this did not lead anywhere. Could so please have a hint?
  32. S

    Length of AD inside a triangle ABC

    I get the answer but my working is really long: 1) Find all the length of sides of the triangle 2) Let DB = x, so CD = CB - x 3) Compare the area of triangle ADC and ABD using formula 1/2 . a . b sin θ then find x 4) Find cosine of angle B by using cosine rule on triangle ABC 5) Use cosine rule...
  33. Narasoma

    I Spacetime Curvature via Triangle

    I understand the mechanism of defining the curvature of a 2D manifold via triangle. But I don't understand how this works in 3D. Meanwhile, Lawrence Krauss mentioned in his book A Universe from Nothing it does. How does this work in 3D?
  34. chwala

    Solving this problem that involves the area of triangle

    I am still looking at this question. One thing that i know is that the distance ##AB=\dfrac {(λ+1)\sqrt {2λ^2-2λ+1}}{λ^2}## distance ##OA=\sqrt 2## distance ##OB=\dfrac{\sqrt{λ^2+1}}{λ^2}## Perpendicular distance from point ##B## to the line ##OA=\dfrac{\sqrt{2(λ^4+2λ^3+2)}}{2λ^2}## Therefore...
  35. S

    Find the minimum perimeter of a triangle with these constraints

    My attempt: $$|\vec{BA} -\lambda \vec {BC}| \geq 2|\vec {BC}|$$ $$|\vec {BA}|^2 -2 \lambda (\vec {BA} \cdot \vec {BC}) +\lambda ^{2} |\vec {BC}|^2 \geq 4|\vec {BC}|^2$$ $$|\vec {BA}|^2 -2 \lambda |\vec {BA}| \cos \theta +\lambda ^{2} \geq 4$$ Am I even on the right track? Thanks
  36. T

    MHB Find this unknown angle in the triangle

    Find the unknown angle in the triangle. It is quite an interesting task. I will put a trigonometric solution, Mabra I wonder how it opens geometrically. Help someone
  37. S

    MHB Can an Ellipse Help Solve the Scalene Triangle Problem?

    Hi guys! I've got a problem with a triangle, and I'm frazzled my brain trying to work it out (not even sure if what I'm looking for is possible with the info I have!). Pic of the offending triangle attached. Basically, I know the length of C and the sum of lengths A and B. Now, if C was...
  38. S

    Volume of solid having triangle base and semicircle slice

    First, I tried to find the equation of line passing through (2, 0) and (0, 3) and I got ##y=3-\frac{3}{2}x## Then I set up equation for the area of one slice, ##A(x)## $$A(x)=\frac{1}{2} \pi r^2$$ $$=\frac{1}{2} \pi \left( \frac{1}{2}y\right)^2$$ $$=\frac{1}{2} \pi...
  39. chwala

    Calculate the area of the triangle- Vector Calculus

    This is the question, Now to my question, supposing the vectors were not given, can we let ##V=\vec {RQ}## and ##W=\vec {RP}##? i tried using this and i was not getting the required area. Thanks...
  40. Anand Sivaram

    I Finding vertex of a 3D Triangle on a Plane

    I came across the following problem and wondering how to solve it. There is a plane n1x + n2y + n3z + n4 = 0 where n1, n2, n3, n4 are known. The triangle is in this plane. We already know the two vertices P1(x1, y1, z1), P2(x2, y2, z2) of the triangle. Now we have to find the third vertex P(x...
  41. Lorena_Santoro

    MHB Can you find the area of the blue triangle in just 3 minutes?

  42. R

    Boats in a triangle colliding after some time

    Assume that three boats, ##B_1##, ##B_2## and ##B_3## travel on a lake with a constant magnitude velocity equal to ##v##. ##B_1## always travels towards ##B_2##, which in turn travels towards ##B_3## which ultimately travels towards ##B_1##. Initially, the boats are at points on the water...
  43. brotherbobby

    A line integral about a closed "unit triangle" around the origin

    Problem statement : As a part of the problem, the diagram shows the contour ##C##above on the left. The contour ##C## is divided into three parts, ##C_1, C_2, C_3## which make up the sides of the right triangle. Required to prove : ##\boxed{\oint_C x^2 y \mathrm{d} s = -\frac{\sqrt{2}}{12}}##...
  44. M

    MHB Inequality in Triangle: Prove $|x^2+y^2+z^2-2(xy+yz+xz)|\le \frac{1}{27}$

    Let $x, y, z$ be length of the side of a triangle such that $\sqrt{x} + \sqrt{y} + \sqrt{z} = 1.$ Prove $|x^{2} + y^{2} + z^{2} - 2\left( xy+yz+xz\right)| \le \frac{1}{27}$.
  45. U

    Engineering Power Triangle and Power factor

    Induction motors: pf = 0.8 triangle: θ = arccos(0.8) = 36.86 degrees Pa = 500/0.8 = 625 VA Pr = sqrt(625^2-500^2) = 375 VAR Synchronous motors: pf = 0.707 triangle: θ = arccos(0.707) = 45 degrees Pa = 500/0.707 = 707 VA Pr = sqrt(707^2-500^2) = 499.85 VAR I am uncertain of how I can...
  46. G

    I Encircled energy for different aperture shapes (circle, triangle, square)

    Hi all, I have a system whereby, there are different aperture shapes which are: circle, triangle, square e.t.c. this apertures are all 300um in diameter. I will like to know if the encircled energy calculated for the different apertures after diffraction will be different due to different...
  47. D

    Simple geometry problem -- Find the perimeter of the triangle ABC

    Hello, so I saw this problem on a website while looking up trigonometric identities and trying to solve it. what I know: The internal angles add up to pi Let the tangent point between A and B be X Let the tangent point between B and C be Y Let the tangent point between C and A be Z ##...
  48. e2m2a

    I Properties of the legs of a right triangle

    I want to know if a right triangle can only have one leg that is a perfect power of a number. Another words is it impossible for a right triangle to have two legs that are numbers that are raised to the same perfect power? Can somebody answer this question and show me the proof?
  49. M

    Combinatorics: Counting quadrilaterals in triangle pattern

    Hi, I was watching a Youtube on combinatorics (here) and a problem was posed at the end of the video about counting the number of quadrilaterals. Question: How many quadrilaterals are present in the following pattern? Attempt: The video started with the simpler problem of finding the number...
  50. M

    Area of interior triangle of pyramid normal to a side length

    This isn't homework, but I figured it's fine if I make it a HW problem and post here (if not, please let me know). Let ##z^*=0## be the vertex of the pyramid, and let ##z^*## run the altitude. It's easy to show the area of the base normal to the altitude is ##A = 4 \left.z^*\right.^2...
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