What is Triangles: Definition and 215 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


{\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. E

    B Pattern inductive reasoning problem

    In attempting to finish the sequence, I realized that the triangles in the inner hexagon is decolored in the direction of the arrow I drew for the given elements of the sequence. Any tips on how to think systematically about similar problems since I am required in the exam to complete like...
  2. Mark Harder

    I Chiral symmetries in E[SUP]^n[/SUP]

    As a biochemist, I deal with chirality of molecules all the time. If you have a tetrahedral molecule, for example a carbon atom, and all 4 vertices are labeled differently, as in different atoms on each one, then that molecule has a mirror-symmetric one that cannot be superimposed on the...
  3. Ahmed1029

    I Triangles on Spheres: Isosceles and Shortest Distance Inferences

    If I have a triangle on a sphere with two of its angles 90 degrees each, do I conclude that it's isosceles and that the shortest distance (on the sphere) beteeen the base and the vertix of the thid angle is 1/4 the circumference of a great circle on the sphere? This is the picture I have in...
  4. Ahmed1029

    I What's the definition of angle in a curved space embedded in a higher Eucledian space?

    I don't want to post this in a math forum because it's very basic and I just want a straightforward answer, not something math heavy . What's the definition of angle in a cuved space embedded in a higher eucledian space? Like when I have a spherical surface in 3d eucledian space and want to work...
  5. B

    MHB Proving Two Triangles are Congruent

  6. Remixex

    Triangles inside a circle to represent raypaths inside an ideal Earth

    I have managed to get some of the required distances and angles. I have the distance ##a##, the velocity inside the mantle, the total radius of the Earth ##R_t## as well as mantle and core radii. I have also figured out the angle of incidence, however I cannot get the refracted angle with the...
  7. M

    MHB Counting Color Combinations in 12 Triangles

    There are 12 triangles (picture). We color each side of the triangle in red, green or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?
  8. N

    MHB Problem about equilateral triangles

    Hi! I need help with this exercise: A side of one equilateral triangle equals the height of a second equilateral triangle. Find the ratio of the perimeter of the larger triangle to that of the smaller. A "detailed solution" to be analysed by me then. Answer: \[ 2/3 sqrt3 \] Thanks
  9. N

    What is the angle needed to solve this right triangle?

    The Figure My Attempt at Solution ##\tan{ACB} = \frac{AB}{BC}, \ \tan41.45^\circ = \frac{AB}{10} \Rightarrow AB = 10\tan45.41^\circ \approx 8.83##cm Similarly ##\tan{CBD} = \frac{CD}{BC}, \ \tan32.73^\circ = \frac{CD}{10} \Rightarrow CD = 10\tan32.73^\circ \approx 6.43##cm After this I...
  10. A

    Perimeter relationships -- Dividing a rectangle into 4 triangles

    Hello, I am studying geometry with an app on my phone. There was a difficult problem, which had two different explanations for solving. I correctly understood one explanation. I reviewed later without memory of the problem at all. There was an obvious attempt from what was learned previously...
  11. K

    MHB Demonstration of a formula for the ratio between the hypotenuses of two triangles

    Hello, everybody: I am a philologist who is fond of mathematics, but who unfortunately has just an elementary high school knowledge of them. I am translating La leçon de Platon, by Dom Néroman (La Bégude de Mazenc, Arma Artis, 2002), which deals with music theory and mathematics in the works of...
  12. anemone

    MHB Triangle Lengths: Can $a,b,c$ Form a Triangle?

    Three positive real numbers $a,\,b$ and $c$ are such that $a^2+5b^2+4c^2-4ab-4bc=0$. Can $a,\,b$ and $c$ be the lengths of the sides of a triangle? Justify your answer.
  13. karush

    MHB -gre.ge.2 distance by similiar triangles

    $\textit{not to scale}$ A summer camp counselor wants to find a length, x, The lengths represented by AB, EB BD,CD on the sketch were determined to be 1800ft, 1400ft, 7000ft, 800 ft respectfully Segments $AC$ and $DE$ intersect at $B$, and $\angle AEB$ and $\angle CDE$ have the same measure...
  14. anemone

    MHB Prove Similar Triangles: $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$

    Prove that two triangles with sides $a,\,b,\,c$ and $a_1,\,b_1,\,c_1$ are similar if and only if $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}=\sqrt{(a+b+c)(a_1+b_1+c_1)}$.
  15. M

    MHB Creating Triangles: How Many Can You Make?

    How many triangles can be made given the following dots? Thanks a lot!
  16. M

    B Find triangles with areas that are integers

    It is pretty obvious that all right-angled triangles whose sides are integers will have areas which are also integers. Since either the base or height will be an even number, half base x height will always come out exactly. However, I have only found one non-right-angled triangle where this is...
  17. S

    I Find Angles in Right Angle Triangles

    If it we know 3 sides of a right angle triangle, will it be possible to calculate the angles without using trig. Functions
  18. M

    MHB Acute angle of right triangles

    Hey! :o We have a rectangle inside a semicircle with radius $1$ : From the midpoint of the one side we draw a line to the opposite vertices and one line to the opposite edge. Are the acute angles of the right triangles all equal to $45^{\circ}$ ? (Wondering) All four triangles are...
  19. karush

    MHB Uncovering the Hidden Simplicity of Similar Triangles

    This seems to be rather common GRE Exam Question Which appears to harder that what it is It looks like a similar triagle solution $\dfrac{800}{7000}=\dfrac{x}{1400}$
  20. jaumzaum

    Is symmetry necessary for solving triple integrals in probability calculations?

    I just started learning triple integrals. I don't know if this is right (I'm only concerned about the limits of the Integral) Consider the case ## a>=b>=c ## $$ P = \lim_{M \rightarrow +\infty} \frac { \int_{a=0}^M \int_{b=\frac a {1.1}}^a \int_{c= \frac a {1.1}}^b \, da \, db \, dc} {...
  21. K

    I Are All Triangles Actually Isosceles? Discover My Greek-Euclidean Proof!

    That is my Greek-Euclidean proof that all triangles are isosceles. Any doubts?
  22. S

    Prove Homothetic Triangles Concurrent: Ceva's Theorem

    Homework Statement Consider the two homothetic non congruent triangles, with their corresponding sides parallel to each other. Prove that extended CE, extended DA and extended BF are concurrent. Homework EquationsThe Attempt at a Solution I can easily solve this using Desargues' theorem. The...
  23. S

    MHB Angles in congruence triangles

    I am trying to work out the above problem. Known are: D,A,B,a,c,d. I need to work out b,d,G and F. Any ideas? c+d = 90 degrees. Regards, Stan
  24. Jim Lundquist

    I Degenerate Triangles: Questions & Poincaré Conjecture

    Please forgive me...I am not a mathematician, but I have a couple questions that have been puzzling me. In theory, can a circle be so large that connecting 3 points on that circle result in a degenerate triangle? If the length of a straight line drawn between two points on a circle is Planck...
  25. A

    Which is harder to pull apart, a circle or a triangle?

    In deciding which shape of ring I should use to secure an anchor to an anchor trolley I came across two choices, a circular ring or a triangular ring. While either will surely work, I began to wonder which would be more difficult to pull apart. Most of the information I found is about forces...
  26. shintashi

    B What is the name of this triangular geometric shape?

    I used to think it was called Zeno's tower, but then realized I probably called it that because it reminded me of his paradox. I have been unable to find this shape on the internet, although I saw a small steel tower outside Stonybrook using this geometry. I have attached an image of the basic...
  27. OldWorldBlues

    B How Can Trigonometry Calculate Distances in Surveying?

    Hi there! I haven't yet taken a trigonometry course (I'm in High-school), but I have an amateur interest in surveying. Recently I began thinking about how I could calculate the height of a point relative to me, or the distance of the object from me. Naturally, I immediately thought of the...
  28. K

    MHB Calculating Coordinates of Spherical Triangles

    Hello, I'm not a student, I'm just trying to figure out how to calculate coordinates on a globe, and I would like to ask for some help. Let's say I have POINT A on the globe with the following coordinates: POINT A Latitude 45° 27' 50.95" N Longitude 9° 11' 23.98" E Also I have POINT B which...
  29. S

    How do I find Vector C in a non-right triangle using the Law of Cosines?

    Homework Statement Vector A: 1.96N at 20° Vector B: 1.71N at 65° There are different parts I did the graph part by hand. I am having trouble finding it analytically. We have to use cm for our triangle. Homework EquationsThe Attempt at a Solution Vector A: 1.96 cm at 20° Vector B: 1.71 cm at...
  30. vantroff

    B (Proof) Two right triangles are congruent.

    Hi, the question is from Serge Lang - Basic mathematics, Page 171 exercise 6. Thing to prove: Let ΔPQM and ΔP'Q'M' be right triangles whose right angles are at Q and Q', respectively. Assume that the corresponding legs have the same length: d(P,Q)=d(P',Q') d(Q,M)=d(Q',M') Then the right...
  31. Z

    Area of Outer Triangle composed of two inner triangles

    Homework Statement 1. [/B]In the figure below, AB=BC=CD. If the area of triangle CDE is 42, what is the area of triangle ADG. See the attached figure Homework Equations I think we can start from area of triangle which is given by: Area of triangle CDE = ½ * CE * DE Or 42 = ½ * CE * DE...
  32. F

    Distance formula vs similar triangles

    Hello, took a year off school, now shaking the rust off. so according to the book using similar triangles d=9.6 I understand how they got the answer, but i used distance formula from the point to the origin and got 9.8 I checked 9.6 and it checks out with the numbers but Idk why using the...
  33. H

    How to find the radii of these 2 circles given 2 known points

    Homework Statement Homework Equations y-y1 = m (x-x1) ---> line equation $$ (x - a)^2 + (y-b)^2 = r^2 $$ ---> circle equationThe Attempt at a Solution I tried to draw the triangles using, (1, 3) (2, 4) and (0, b) (0, b) is the tangent point to y-axisand used those points for making...
  34. D

    I Integer Cevians - Equilateral Triangles

    Is anybody familiar with any theory of integer cevians on equilateral triangles? More specificaly, I was trying to find something about the number of integer cevians that divide the side in integer parts. Like, the eq triangle of side 8 have cevian 7 dividing one side into 3+5. Only reference...
  35. Wrichik Basu

    A problem in Trigonometry (Properties of Triangles) v3

    Homework Statement In any triangle ABC, prove that $$a^2 b^2 c^2 \left (\sin {2A} +\sin {2B} + \sin {2C} \right) = 32 \Delta ^3$$ Here ##\Delta ## means the area of the triangle.Homework Equations The Attempt at a Solution
  36. Wrichik Basu

    A problem in Trigonometry (Properties of Triangles) v2

    Homework Statement In any triangle ABC, prove that $$ a^2 + b^2 +c^2 =4 \Delta (\cot {A}+\cot {B}+\cot {C}) $$ Homework Equations The Attempt at a Solution
  37. Wrichik Basu

    A problem in Trigonometry (Properties of Triangles)

    In any triangle ABC, prove that $$(b+c-a) \left( \cot {\frac {B}{2}} + \cot {\frac {C}{2}} \right)=2a \cot {\frac {A}{2}} $$
  38. kaliprasad

    MHB Infinite Isosceles Triangles w/ Integer Sides & Areas

    Show that there are infiinite isosceles triangles which have integer sides and integer areas
  39. Bunny-chan

    Basic trigonometry in dynamics force problem

    Homework Statement Two spheres, with 0.5g each, are hanging by 30cm threads, tied on the same spot. The same electric charge is communicated to each sphere; in consequence, the threads move apart until they are about 60^\circ from each other. What is the value of the charge? \theta =...
  40. Const@ntine

    B How do I determine angle equality in this graph?

    Okay, this might be a tad silly, but I cam across this graph/picture i my physics textbook, and for the life of me I can't figure out how to connect the two angles. Here, have a look: It's not a big part of what the paragraph is about, so in theory I could just skip it, but it's been...
  41. C

    MHB Are the coordinates of the vertices of this triangle all integers?

    L1 [x,y]=[2,1]+r[-5,1] L2 [x,y]=[1,4]+s[2,1] L3 [x,y]=[3,5]+t[4,-5] These three lines are sides of a triangle find: 1)the perimeter of the triangle 2) The largest angle 3) the centroid of the triangle so I converted the vector equations into parametric, and then made two of the x parametric...
  42. Albert1

    MHB How many triangles can be proclaimed as right angled ?

    24 points $A_1,A_2,-----,A_{24}$ equally divide the circumference of circle $O$ ,any three of the 24 points will determine an inscribed triangle, now how many triangles can be proclaimed as right angled ?
  43. J

    Structural Support Triangles for a rope-climbing robot

    sorry if this is in the wrong spot or what not So my school has this robotics club and we need to build a robot to climb a rope as well as do other things. But focusing on the rope climbing I need to design 2 plates that can support the weight of the robot (~150lbs) The plates themselves are...
  44. J

    Finding the center of mass of an arbitrary uniform triangle

    Homework Statement 1. Show that for an arbitrary uniform triangle ABC, with A at (x1, y1), B at (x2,y2), C at (x3, y3), the CM (xcm, ycm), is simply defined by xcm=(x1+x2+x3)/3, and ycm =(y1+y2+y3)/3 Homework Equations xcm = 1/M * ∫xdm ycm = 1/M * ∫ydm M = ∫dm = ∫δdA where δ = M/A = dm/dA...
  45. M

    Electrostatic Forces in Equilateral Triangles

    Homework Statement I've encountered a question about electrostatic forces on vertices of an equilateral triangle and I believe that I solved it correctly but my Physics teacher has marked it as incorrect. Am I correct? Amy clue why my Physics teacher marked it wrong? This is the question and...
  46. T

    I Solving For Right Triangles With A Twist

    Hello Community Firstly this is not a homework question despite the problem appearing to be very homework like. I work in engineering. I am trying to determine the amount of clearance I have from a point on the ground to a wall. I have two points, Point A and Point B. Point A is 30mm Higher...
  47. CollinsArg

    B Trying to represent a written geometrical description

    I'm reading an old book about Thales (Greek geometry), and I can't understand what the next part means, and how to represent it graphically, could you help me? thanks: It begin stating that if you divide an equilateral traingle with a perpendicular from a vertex on the opposite side, it'll be...
  48. M

    MHB Ratio of the area of triangles

    In the figure , the area of triangle $ABC$ is twice that of triangle $BCD$.USing the given information , find the ration of the area of the triangle $CFG$ to the area of triangle $BEG$ Hint- Use the midpoint theorem. (Wave) Stuck in this problem & currently I have no workings to show.
  49. M

    MHB Show that AN=1 cm using the knowledge of equiangular triangles.

    I need help with the fourth part of this series of the questions, Here's the copied figure ii. AM=MB (M midpoint) NAM=MBP($90^\circ$) AMN=BMP(vertically opposite) $\therefore \triangle AMN \cong \triangle BMP \left(AAS\right)$iii.MN=MP (triangle AMN & triangle BMP congruent) MC=MC (Common...