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 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 $P$.
  2. 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...
  3. L

    MHB Max and min tangent and triangle

    in what point of the circumference: x2 + y2 = 1 the tangent to this, (to yhe circunference) form with the coordinate axes the triangle of smaller area?. answer ( +/-(sqrt2/2), +/-(sqrt2/2) ) Ok y2= 1-x2 Now I don't know in what point must i get the tangent?? I don't think it is in 0,0 I...
  4. K

    MHB Trigonometry and Triangle Questions

    Gents, Thanks for all the guidance in the past with my work and brothers too. Can anyone help with these , see file attached
  5. S

    How Do You Calculate Triangle Peak Equations in MATLAB?

    Hi, I am working on some MATLAB for my engineering classes and I need help with understanding a given diagram. In the attached diagram, I need help understanding how we determined the equations For example, I would like to know the equations for a triangle with the peak at 2/3L I know...
  6. P

    Coulomb's Forces triangle of charges

    Homework Statement Consider the triangle of charges diagrammed below, for which d = 5 cm, q = 2 nC, and the +x-axis points to the right. What is the force Fvec on the 1 nC charge? Give your answer as a magnitude and a direction.Homework Equations F=K(q)(Q)/r^2 The Attempt at a Solution I...
  7. B

    Calculating Fluid Force of Triangle Submerged in Water

    Homework Statement Find the fluid force of a triangle with base of 6 ft and height of 4 ft submerged vertically into a body of water,vertex down, to a depth of 3 feet. Homework Equations from c to d W ∫ h(y) L(y) The Attempt at a Solution I set up the triangle on an x and...
  8. E

    How Does Charge Distribution Affect Electric Field and Potential at Point P?

    Homework Statement Two charges of opposite sign and equal magnitude Q = 0.82 C are held 2.0 m apart as shown in the figure. Picture: http://gyazo.com/a78a049ce0a24ad464fd7bb99dffd665 A) Determine the magnitude of the electric field at the point P. B) Determine the electric potential...
  9. Saitama

    MHB Triangle and logarithm problem

    Problem: Given that $a,b$ and $c$ are the sides of $\Delta ABC$ such that $$z=\log_{2^a+2^{-a}} \left(4(ab+bc+ca)-(a+b+c)^2\right)$$ then $z$ has a real value if and only if A)a=b=2c B)3a=2b=c C)a-b=3c D)None of these Attempt: I am not sure where to start with this kind of problem. I wrote the...
  10. craigi

    Many Worlds, Indistinguishability, Pascal's Triangle and the Gaussian

    Suppose we have a quantity which can take discrete equally spaced values. Iteratively, we can increase or decrease this quantity by one quantum, splitting into two new worlds each time. After multiple iterations we have some indistinguishable worlds, as described by Pascal's triangle. As the...
  11. M

    Why can't we apply trig ratios to any angle of a right triangle?

    Why do trigonometric ratios have to be related to the angle between the base and hypotenuse of a right angle triangle? I am trying to understand why I can't use these ratios to any angle of a right angle triangle. I try to do that in the attached document. It seems to work for all ratios...
  12. E

    Difference between Force components using triangle rule and dotproduct

    Hello everyone, I have here one important abstract question which makes up some perplexity to my understanding. Attached to this post is one pic of F and two new introduced axes ( u and v ) . Let us for instance not consider them as perpendicular to each other. Now, if I...
  13. B

    Coulomb's Law (3 charges in equilateral triangle)

    Three charged particles are placed at the corners of an equilateral triangle of side L = 1.74 m. The charges are q1 = 3.63 µC, q2 = −8.05 µC, and q3 = −6.31 µC. Calculate the magnitude and direction (counterclockwise from the positive x axis) of the net force on q1 due to the other two...
  14. J

    MHB Problem solving 2 variable equal triangle problem

    I solved this many years ago, but after revisiting Trig in order to tutor my daughter, I revisited this to stimulate myself but am hitting a brick wall. Problem: A 4 inch square sits in a corner(picture x,y origin). A 12 inch ruler or line leans against the wall at an angle such that there are...
  15. anemone

    MHB Find the area of an equilateral triangle

    Show that the curve $x^3+3xy+y^3=1$ has only one set of three distinct points, $P$, $Q$, and $R$ which are the vertices of an equilateral triangle, and find its area.
  16. I

    Understanding Power Triangle and Box Conversions: Explained and Simplified

    I understand that S = P + jQ however I am confused how they got from that to the yellow box. Also, how did they get from the yellow box to the pink box and from the pink box to the blue box? Thanks
  17. P

    Effect of pointcharges on electric dipole in an equilateral triangle

    2 particles of charge q are placed at 2 vertices of an equilateral triangle of side a. An electric dipole is placed at the third vertex with its dipole moment orientated parallel to the opposite side of the triangle. a) Determine the magnitude of the torque on the dipole due to the electric...
  18. L

    MHB Maximizing the volume of a cone formed by revolving a right triangle

    A triangle hypotenuse given rectangle is rotated around one of their legs to generate a right circular cone? find the cone of greater volume. resp V= (2Sqrt(3)pi L^3)/27 It says hypotenuse given but it has no value According to the answer you can name it L
  19. anemone

    MHB Proving Triangle Angles Relation: $\frac{1}{\sin^2 \lambda}$

    Inside a triangle $ABC$, there is a point P satisfies $\angle PAB=\angle PBC=\angle PCA=\lambda$. If the angles of the triangle are denoted by $\alpha$, $\beta$ and $\gamma$, prove that $\dfrac{1}{\sin^2 \lambda}=\dfrac{1}{\sin^2 \alpha}+\dfrac{1}{\sin^2 \beta}+\dfrac{1}{\sin^2 \gamma}$
  20. U

    Convolution, Triangle Function

    Homework Statement Part (a): Find the intensity as function of ##\theta## and sketch it. Part (b): Find the intensity as function of ##\theta## and sketch it. Comment on first minima. Homework Equations The Attempt at a Solution Part(a) Convolution Method V_b = \frac{1}{2a}, 0 \leq...
  21. D

    Medians of an Isosceles triangle

    Homework Statement Prove that the medians to the equal sides of an isosceles triangle divide each other into respectively equal parts Homework Equations The Attempt at a Solution suppose we have a triangle ABC where AB = AC. Let D be the point on AB in which the median intersects...
  22. C

    Triangle inequality proof in Spivak's calculus

    So hi, there's one little thing which I'm not understanding in the proof. After the inequality Spivak considers the two expressions to be equal. Why?!? I just don't see why we can't continue with the inequality and when we have factorized the identity to (|a|+|b|)^2 we can just replace...
  23. U

    Find Type of Triangle Given r1-r2-r3

    Homework Statement If in any triangle r=r1-r2-r3, then which type of triangle is it? r1,r2,r3 denotes ex-radii. The Attempt at a Solution 4R = r1+r2+r3-r r2+r3 = 2R I can't figure out what to do next.
  24. U

    Determine which triangle is this

    Homework Statement If in a ΔABC , \left( 1-\dfrac{r_1}{r_2} \right) \left( 1-\dfrac{r_1}{r_3} \right) = 2 then which triangle is it? r1, r2, r3 denotes ex-radii. The Attempt at a Solution I get this condition a^2+2b^2+2c^2+3bc=3ac+3ab Also \sum tan \frac{A}{2} tan \frac{B}{2} = tan^2...
  25. anemone

    MHB Finding $\angle MBC$ in Triangle $ABC$ with Proof

    Given a triangle $ABC$ such that $\angle BAC=103^{\circ}$ and $\angle ABC=51^{\circ}$. Let $M$ be a point inside triangle $ABC$ such that $\angle MAC=30^{\circ}$, $\angle MCA=13^{\circ}$. Find $\angle MBC$ with proof.
  26. G

    Proof of Excircle at Triangle's Intersection Pt

    This is just a curiosity from my part: Has anyone know the proof of "Intersection of one internal angle bisector and two external angle bisectors of triangle is the center of an excircle."? I tried some things, but no luck.
  27. N

    Length change of diagonal in deformable triangle

    Hi I have a problem understanding a figure in my lecture notes. The figure is the following one It shows the deformation of a triangular element from time t to time t+dt: So at t it is a isosceles triangle and at t+dt it is deformed. According to my lecture notes (page 16, eq. 28), the...
  28. T

    MHB Solve Triangle Height without Trig - Help for Kids

    My daughter was given this problem in an exam. She could not do it and was never given the solution. I can do the problem using trig but she has not been taught trig yet. She says that it can be done without trig using a more simple method that she doesn't know. Could somebody please explain...
  29. G

    Optimization - rectangle inscribed in a right triangle

    Homework Statement A rectangle is to be inscribed in a right triangle having sides 3 cm, 4 cm and 5 cm, as shown on the diagram. Find the dimensions of the rectangle with greatest possible area. Homework Equations 1. x^{2}+y^{2}=w^{2} in terms of w=\sqrt{x^{2}+y^{2}} 2...
  30. C

    Is There a Mistake in Solving this Right Angle Triangle Problem?

    The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other; express both the angles in degrees. So I have found the following answers : 810/17=47,05... degrees and 810/17=47,05... grades which gives 42,35... degrees Now, the real answer...
  31. C

    Equilateral triangle net force

    Homework Statement Lets say I have an equilateral triangle, and you are asked to calculate the net force on the top of the triangle Homework Equations Coulombs Law The Attempt at a Solution I know to use Coulombs law to find the forces acting on the top of the triangle, then since...
  32. MarkFL

    MHB Prove Triangle Inscribed in Semicircle Is Right Angle | Arundev Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  33. Albert1

    MHB Triangle Inequality Proof for Side Lengths of Triangle ABC

    Triangle ABC with side lengths a,b,c please prove : $ \sqrt {ab}+\sqrt {bc}+\sqrt {ca}\leq a+b+c<2\sqrt {ab}+2\sqrt {bc}+2\sqrt {ca}$
  34. P

    Derivatives, rates of change (triangle)

    1. A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is \frac{\pi}{3}, this angle is decreasing at a rate of -\frac{\pi}{3} rad/min. How fast is the plane traveling at that time? Homework Equations...
  35. C

    How to determine angle of net force-electric charges of right triangle

    Homework Statement The .274 and .88 was found using the equation of electrostatic force . Homework Equations K=q1q2/d^2 The Attempt at a Solution Would I just tan inverse of .274N and .88N which would be 17 degrees. However my question is: how to determine the reference pt: would...
  36. M

    Moment of inertia of triangle about centroidal axis

    Homework Statement There is the moment of inertia about an x and a y-axis named, I_{x}, I_{y}. Then there is the moment of inertia about the centroidal x and y-axis named, \overline{I}_{x}, \overline{I}_{y}. Often we can look up these values in a table (like the figure included) and apply...
  37. MarkFL

    MHB Kylie's question at Yahoo Answers regarding minimizing the area of a triangle

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  38. K

    Alternative way to calculate the area of a right angled triangle

    Hello all! New to the forums, and I have a question for you. In my classes, we have been dealing a lot with proofs lately, so when I was working on an assignment, I figured I would try and find my own proof for something, just for the hell of it. I decided to tacle the area of a right angled...
  39. V

    Max Area of Triangle with Sides (0,1], [1,2], [2,3]

    Homework Statement Find the maximum area of a triangle with sides a\in (0,1] ,b\in [1,2], c\in [2,3]. Homework Equations The Attempt at a Solution I tried to make the area as a function of a single variable so that by differentiating I can get the answer. But it was...
  40. anemone

    MHB Proving Area Ratio of Equilateral Triangle Divided by Line

    A line divides an equilateral triangle into two parts with the same perimeter and having areas $A_1$ and $A_2$ respectively. Prove that $\dfrac{7}{9} \le \dfrac{A_1}{A_2} \le \dfrac{9}{7}$.
  41. U

    Triangle with median and altitude

    Homework Statement In triangle ABC with A as obtuse angle, AD and AE are median and altitude respectively. If BAD = DAE=EAC, then sin^3(A/3)cos(A/3) equals Homework Equations The Attempt at a Solution CE = a/2. Let DE = x. Then BD = a/2 - x. Let AE = p, AD = q. For ΔADB \cos...
  42. T

    Solving for base angles in a congruent triangle

    Homework Statement Solving for base angles in a congruent triangle One of the base angles in a congruent triangle is 3x+5 The other base angle is unknown The remaining angle is x+16. Homework Equations See attached PDF The Attempt at a Solution Because two base angles in a...
  43. T

    Triangle Inequality in 'Linear Algebra Done Right'

    I'm stuck on one aspect of the proof on page 105 of the 2nd edition. Equation 6.13 is necessary for the inequality to be an equality as it says but they never seem to account for inequality 6.11. Specifically, I don't see how this satisfies 2 Re<u,v> = 2 |<u,v>| Thanks for any guidance.
  44. N

    Normal Modes of a Triangle Shaped Molecule

    Homework Statement A molecule consists of three identical atoms located at the vertices of a 45 degree right triangle. Each pair of atoms interacts by an effective spring potential, with all spring constants equal to k. Consider only planar motion of this molecule. What are 6 normal modes and...
  45. A

    Explain hamming distance satisfies the triangle inequity

    Hamming distance satisfies the triangle inequity that is for all x, y, u in c such that d(x,y) <= d(x,u) + d(u,y) where c is a code. Also when does the equality hold? My approach is Turning x to u by changing at most d(x,u) letters and turning u to y by changing at most d(u,y) letters. So...
  46. T

    Unknown lengths of triangle, known area and internal angles

    Homework Statement I know the area of a right angled triangle, I also know the ratio of the two non-hypotenuse sides. Is there anyway of finding the lengths? Thanks! Homework Equations I don't think there are any like Area = 1/2abSinC The Attempt at a Solution I obviously...
  47. D

    What is the center of mass in a triangular beam?

    Hello,, I'm reading about beams in mechanics of matrials. But I'm a bit stuck because of one thing that I either don't understand or just don't comply with! So, here's a picture from the book: picture 003 It says that the center of mass should be at 2/3 of the length.. I tried to...
  48. Saitama

    Triangle problem - Evaluating the given expression

    Homework Statement In a triangle ABC, with usual notation, if ##a^2b^2c^2 (\sin 2A + \sin 2B + \sin 2C) = λ(∆)^x## where ∆ is the area of the triangle and x ##\in## Q, find (λx). Homework Equations The Attempt at a Solution The usual notation is: a,b,c are three sides of the...
  49. karush

    MHB How fast is the area of triangle formed

    9 A Ladder 13ft long is leaning against the side of a building. If the foot of the ladder is pulled away from the building at a constant rate of 8in per second how fast is the area of triangle formed by the ladder, the building and the ground changing (in feet squared per second) at the instant...
  50. A

    Real Analysis triangle inequity

    Prove |x|+|y| ≤ |x+y| + |x-y| for all real numbers x and y. Some ideas I have is let a = x+y and b = x-y and apply triangle inequity Could anyone give me some direction? Thanks
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