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. N

    Perpendicular Bisector of a triangle

    Here is my attempt to draw a diagram for this problem: I'm confused about the "the perpendicular bisector of ##BC## cuts ##BA##, ##CA## produced at ##P, \ Q##" part of the problem. How does perpendicular bisector of ##BC## cut the side ##CA##?
  2. anemone

    MHB Triangle Equality Proof: $\frac{a}{b}=1+\sqrt{2\left(\frac{c}{b}\right)^2-1}$

    Let $ABC$ be a triangle with $\angle ACB=2\alpha,\, \angle ABC=3\alpha$, $AD$ is an altitude and $AE$ is a median such that $\angle DAE=\alpha$. If $BC=a,\,CA=b$ and $AB=c$, prove that $\dfrac{a}{b}=1+\sqrt{2\left(\dfrac{c}{b}\right)^2-1}$.
  3. B

    How to write basic syntax for triangle recursion?

    Hi, I'm new to programming in python [total beginner in programming] and I would like to ask you for your help. Here is what I got so far: import numpy as np import random from math import sqrt p = np.array([(0, 0), (1, 0), (1, (1/sqrt(2)))], dtype=float) t = np.array((0, 0)...
  4. Who_w

    Moment of inertia of a regular triangle

    Please, I need help! I need to calculate the moment of inertia of a triangle relatively OY. I have an idea to split my triangle into rods and use Huygens-Steiner theorem, but after discussed this exercise with my friend, I have a question: which of these splits are right (picture 1 and 2)? Or...
  5. P

    Solving a Vector Triangle Differential Equation

    By considering a vector triangle at any point on its circular path, at angle theta from the x -axis, We can obtain that: (rw)^2 + (kV)^2 - 2(rw)(kV)cos(90 + theta) = V^2 This can be rearranged to get: (r thetadot)^2 + (kV)^2 + 2 (r* thetadot)(kV)sin theta = V^2. I know that I must somehow...
  6. K

    MHB Is triangle ANC and isosceles triangle and why

    is this triangle isosceles A(7,-2) B(-1,-8) C(5,0)
  7. anemone

    MHB Proving Sum of Tangents $\ge$ Sum of Cotangents in Acute Triangle

    In an acute triangle $ABC$, prove that $\sqrt{\tan A}+\sqrt{\tan B}+\sqrt{\tan C} \ge \sqrt{\cot \left(\dfrac{A}{2}\right)}+\sqrt{\cot \left(\dfrac{B}{2}\right)}+\sqrt{\cot \left(\dfrac{C}{2}\right)}$.
  8. Baluncore

    I What is the name of the triangle centre point with 120° subtended vertices?

    I am looking for the name of the triangle centre point from which the vertices subtend 120°. I am concerned really with deviant equilateral triangles, thru to right angle triangles. Flat triangles, that have an internal angle greater than 120°, do not have such a point within the triangle...
  9. S

    MHB Proof of Triangle Inequality for $n$ Natural Numbers

    Prove for all $n\in N$ $\dfrac{|a_1+...a_n|}{1+|a_1+...+a_n|}\leq\dfrac{|a_1|}{1+|a_1|}+...\dfrac{|a_n|}{1+|a_n|}$
  10. A

    MHB Calculating Perimeter & Area of a Parallelogram & Triangle

    Find the perimeter and area of CD, if ABCE is a parallelogram and ADE is an equilateral triangle.
  11. anemone

    MHB What is the Minimal Area of a Right-Angled Triangle with an Inradius of 1 Unit?

    What is the minimal area of a right-angled triangle whose inradius is 1 unit?
  12. M

    MHB Area of Triangle Shaded Region

    Hello everyone. I am having trouble finding the area of the shaded region using the determinant area formula. I know where to plug in the numbers into the formula. My problem here is finding the needed points in the form (x, y) from the given picture for question 21.
  13. anemone

    MHB Prove Triangle Sides Inequality $\sqrt{b+c-a} \dots \le 3$

    Let $a,\,b,\,c$ be the sides of a triangle. Prove that $\dfrac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\dfrac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\dfrac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\le 3$.
  14. A

    MHB Proving Triangle Area ≤ $\frac{1}{2}$ in a Square with $(n+1)^2$ Points

    Consider a square with the side of length n and $(n+1)^2$ points inside it. Show that we can choose 3 of them to determine a triangle (possibly degenerate) of area at most $\frac{1}{2}$. I think that I know how to solve the problem for the cases $n=1$ and $n=2$: For $n=1$ we can easily prove...
  15. O

    B Paradox of Simultaneous Flashing in a Moving Triangle of Lights

    There are 3 lights in the form of a triangle... A, B, and C are lights and are stationary with respect to each other. S1, S2, S3 are spaceships. B S1 S2 A S3...
  16. karush

    MHB Act.ge.5 third angle of triangle

    $\tiny{act.ge.5}$ ok you have 2 seconds to figure this one out:unsure: This question has live answer choices. Select all the answer choices that apply. The correct answer to a question of this type could consist of as few as one, or as many as all five of the answer choices. \item In triangle...
  17. V

    MHB Minimum Length of Longest Side in Inscribed Triangle

    In triangle ABC, ∠C = 90 degrees, ∠A = 30 degrees and BC = 1. Find the minimum length of the longest side of a triangle inscribed in triangle ABC (that is, one such that each side of ABC contains a different vertex of the triangle).
  18. anemone

    MHB Prove Triangle Inequality: $\sqrt{2}\sin A-2\sin B+\sin C=0$

    In a triangle $ABC$ with $\sqrt{2}\sin A-2\sin B+\sin C=0$, prove that $\dfrac{3}{\sin A}+\dfrac{\sqrt{2}}{\sin C}\ge 2(\sqrt{3}+1)$.
  19. anemone

    MHB Proving $\dfrac{3}{2}$ Inequality in Acute Triangle $ABC$

    In an acute triangle $ABC$, prove that $\dfrac{\cos A}{\cos (B-C)}+\dfrac{\cos B}{\cos (C-A)}+\dfrac{\cos C}{\cos (A-B)}\ge \dfrac{3}{2}$.
  20. anemone

    MHB Proving Angles in Triangle ABC < 120° & Cos + Sin > -√3/3

    All the angles in triangle $ABC$ are less than $120^{\circ}$. Prove that $\dfrac{\cos A+\cos B+\cos C}{\sin A+\sin B+\sin C}>-\dfrac{\sqrt{3}}{3}$.
  21. anemone

    MHB Find maximum area of a triangle

    Given two moving points $A(x_1,\,y_1)$ and $B(x_2,\,y_2)$ on parabola curve $y^2=6x$ with $x_1+x_2=4$ and $x_1\ne x_2$ and the perpendicular bisector of segment $AB$ intersects $x$-axis at point $C$. Find the maximum area of $\triangle ABC$.
  22. P

    Finding the generator of rotations for a 3-state triangle

    I first computed the operator ##\hat{T}## in the ##a,b,c## basis (assuming ##a = (1 \ 0 \ 0 )^{T} , b = (0 \ 1 \ 0)^{T}## and ##c = (0 \ 0 \ 1)^{T}##) and found $$ \hat{T} = \begin{pmatrix} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{pmatrix}.$$ The eigenvalues and eigenvectors corresponding to this matrix...
  23. TytoAlba95

    Find the third side of a Triangle

    Where did I go wrong?
  24. P

    Understanding a Velocity-Time Graph

    Summary:: I think we are still in the earlier parts of Physics and I am confused at how "values" work for a velocity-time graph. We are using the formulas to solve an area of a triangle and rectangle to find the total displacement. If a diagonal line begins from above and continue to go down...
  25. G

    To find the magnitude of the resultant of forces around a triangle

    Could I please ask for advice with the following: ABC is a right-angled triangle in which AB = 4a; BC = 3a. Forces of magnitudes P, Q and R act along the directed sides AB, BC and CA respectively. a) Find the ratios P:Q:R if their resultant is a couple. b) If the force along the directed line...
  26. G

    To find the ratio of forces around a right triangle

    Could I please ask for help with the following: ABC is a right-angled triangle in which AB = 4a; BC = 3a. Forces of magnitudes P, Q and R act along the directed sides AB, BC and CA respectively. Find the ratios P:Q:R if their resultant is a couple. Book answer is 4 : 3 : 5 Here's my diagram...
  27. greg_rack

    Radius of circumference circumscribed to a triangle

    At first I had no idea of how to solve this problem, but checking online I found out that there is a formula linking the radius of the circumference and the side of the triangle... the formula is: side=radius√3 The thing is that I can't understand why is this working... which deduction have been...
  28. E

    Meaning of small triangle in chemical potential

    It is given that the solution is ideal, i.e. that we can take ##\gamma_A = 1##. I wondered what that small triangle signifies in the second definition? Thanks!
  29. cwill53

    Zero Force from 3 charges placed at the vertices of a Triangle

    I'm not really sure where to start with this problem, but I wanted to ask a few questions about the approach I should use. Is it reasonable to say that a gradient could be set up that could describe the force on the fourth ion at any point? The way I'm thinking of this problem is, I want to...
  30. LCSphysicist

    Three charges attached to the vertices of a triangle

    Actually we find the two position in the axis very easily, but, what am trying to find is if exist such position (Being the charge of the ions equal) away from the symmetry axis, but i really don't want to try find it numerically, it would be a disaster. The only conclusion i got is, if such...
  31. Nexus99

    Moment of inertia of an isosceles triangle

    I did in this way: ## I = \int dm \rho^2 ## Dividing the triangle in small rectangles with ##dA = dy x(y) ## where ##x(y) = 2 ctg( \alpha ) (h - y) ## we have : ## dm = \sigma 2 ctg( \alpha ) (h - y) ## Now i have ## \rho^2 = x^2 + (h-y)^2 ## Now I don't know what I can do because it would be...
  32. anemone

    MHB What is the area of triangle $STV$?

    Hi all, I happened to see this primary 6 math geometry problem and thought it was a fun (not straightforward but not too hard) problem. Try it and post your solution if you are interested. (Cool) In the figure, not drawn to scale, $UX=XY=YT$ and $UV=VS$. Given that the area of triangle $XVU$ is...
  33. J

    MHB Solve Right Triangle Problem Without Knowing Bottom Line

    How would I have to calculate this question for an answer, a friend of mine told me he could get the answer without knowing that the bottom line was 16 meters, I can't seem to find a way that would work, I am not sure if I am missing something or he is lying.
  34. anemone

    MHB Prove Triangle Inequality: $\frac{a}{\sqrt[3]{4b^3+4c^3}}+...<2$

    Let $a,\,b$ and $c$ be the side lengths of a triangle. Prove that $\dfrac{a}{\sqrt[3]{4b^3+4c^3}}+\dfrac{c}{\sqrt[3]{4a^3+4b^3}}+\dfrac{a}{\sqrt[3]{4b^3+4c^3}}<2$.
  35. xyz_1965

    MHB What are the angles of isosceles triangle ABC?

    In isosceles triangle ABC, the sides are of length AC = BC = 8 and AB = 4. Find the angles of the triangle . Express the answers both in radians, rounded to two decimal places, and in degrees, rounded to one decimal place. Solution: I think dropping a perpendicular line from the vertex at C...
  36. xyz_1965

    MHB How can I find the area of a triangle with a given angle and two sides?

    Find the area of a triangle with angle 70° in between sides 6 cm and 4 cm. Solution: From the SOH-CAH-TOA mnemonic, I want the ratio of the opposite side (CD) to the hypotenuse (AC). I should be using the *sine* function, not cosine. Yes? SOH leads to sin = opp/hyp sin(70°) = CD/4 CD = 4...
  37. xyz_1965

    MHB Finding the Leg Lengths of a Right Triangle with an Acute Angle of 22°

    A right triangle has an acute angle measure of 22°. Which two numbers could represent the lengths of the legs of this triangle? OPTIONS a. 2 and 5 b. 1 and 5 c. 3 and 5 d. 4 and 5 I know that each leg represents the sides of the right triangle opposite the hypotenuse. I think the tangent...
  38. S

    Using a determinant to find the area of the triangle (deriving the formula)

    This is the question. The following is the solutions I found: I understand that the first line was derived by setting one vertex on origin and taking the transpose of the matrix. However, I cannot understand where the extra row and column came from in the second line. Can anyone explain how...
  39. anemone

    MHB Prove Triangle ABC is a Right Triangle

    Suppose the lengths of the three sides of $\triangle ABC$ are integers and the inradius of the triangle is 1. Prove that the triangle is a right triangle.
  40. anemone

    MHB Area Triangle: Find Formula in Terms of $p, q, r$

    Find in terms of $p,\,q$ and $r$, a formula for the area of a triangle whose vertices are the roots of $x^3-px^2+qx-r=0$ in the complex plane.
  41. anemone

    MHB Can You Solve the Triangle Sides Challenge?

    It is given that the ratio of angles $A,\,B$ and $C$ is $1:2:4$ in a $\triangle ABC$, prove that $(a^2-b^2)(b^2-c^2)(c^2-a^2)=(abc)^2$.
  42. anemone

    MHB Prove ABC is an equilateral triangle

    Prove $ABC$ is an equilateral triangle if $\dfrac{\cos A+\cos B+\cos C}{\sin A+\sin B+\sin C}=3\cot A \cot B \cot C$.
  43. D

    Triangle calculation for the resultant velocity

    Top example- How do I get to 31.7 m/s from 30.8 and 7.7? This is way over my head and need help. Thanks in advance Dean
  44. anemone

    MHB What are the angles of an isosceles triangle with a specific ratio?

    Let $ABC$ be an isosceles triangle such that $AB=AC$. Find the angles of $\triangle ABC$ if $\dfrac{AB}{BC}=1+2\cos\dfrac{2\pi}{7}$.
  45. anemone

    MHB Proving Equality of Angles in an Acute Triangle

    Given that $ABC$ is an acute triangle with $AC>AB$ and $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside the triangle $ABC$ such that $PD$ is parallel to $AE$ and $\angle BAP=\angle CAE$. Prove that $\angle ABP=\angle...
  46. anemone

    MHB Length of Shortest Side In a Triangle

    If $a,\,b$ and $c$ are the sides of a triangle $ABC$, prove that if $a^2+b^2>5c^2$, then $c$ is the length of the shortest side.
  47. C

    MHB Using laws of sines and cosines to solve for a triangle

    Hello, I have problem with this task, I must solve the triangel if i know a:b=2:3, c=15 cm, alfa:beta=1:2.Can you help me please?If you know it write me way how you solve it Thank so much.
  48. V

    Calculate side DC in this triangle

    Was just wondering if someone could take a look at my calculations and see if I've done them correctly.
  49. M

    How Do You Find the Line of Symmetry for Isosceles Triangle ABC?

    ABC is an isosceles triangle such that AB=AC A has coordinates (4,37) B and C lie on the line with equation 3y=2x+12 Find an equation of...
  50. Rongeet Banerjee

    Moment of Inertia for a Triangle with Masses at the Vertices

    If I take the three masses individually and try to calculate the moment of inertia of the system separately then I=(m*0²)+(m*(l/2)²)+(m*l²) =ml²/4 +ml²=(5/4)ml² But If I try to calculate Moment of Inertia of the system using its Centre of mass then As centre of mass is located at the the...
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