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. O

    Rate of increase of a side of a triangle

    Hello, As I believe is normal policy, please don't give me the answer to the problem :) I would, however, greatly appreciate a bump in the right direction. Homework Statement Triangle abc: side a=12, side b=15, angle C is increasing at 2 degress/minute. How fast is the length of side...
  2. G

    Forces in the triangle and pulley system

    Two blocks are connected by a string-one on an incline, the rope draped over a pulley, and one hanging from the string. The smooth inclined surface makes an angle of 35 with the horizontal, and the block on the incline has a mass of 5.7 kg. The mass of the hanging block is 3.2kg. Find the...
  3. D

    Triangle and two circles theorem

    Can someone help me understand why this is the case? I tried forming a cartesian equation for the two circles but there were too many variables that it would be too messy to compute. Otherwise I am rather stuck on how to do it. I would appreciate it if someone can explain how to prove this...
  4. S

    How to Find λ in a Triangle with Given Vectors OA, OB, P, and Q?

    Homework Statement Given that: OA = 5s OB = 10 t P is on the middle of OA and AQ = 3 QB Find λ if BG = λ BP Homework Equations basic vector The Attempt at a Solution I have found AB, BQ, OQ, BP; all in terms of s and t. How to find λ? I've tried but always came back to...
  5. B

    Find max area of rectangle inside a right triangle.

    Homework Statement Find the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of lengths 5cm and 12cm. the two sides of the rectangle lie along the legs. Homework Equations N/a The Attempt at a Solution
  6. A

    Finding Center of Mass of an Isosceles Triangle

    Homework Statement an isosceles triangle with mass m and constant density is placed on an xy plane with base on the y axis. its height ranges from (0,-b/2) to (0,b/2) and its height is (a,0) Find its center of mass. I know that its density is equal 2m/ab. The Attempt at a Solution...
  7. C

    Triangle Cantilever Truss Design

    1. I am an architecture student and the assignment is to design a truss, set up the problem and solve it using graphic statics. I think I did it correctly using the graphical method but I am interested in solving it mathematically. Did I set the problem up correctly in my diagram and if so...
  8. S

    Proving the Triangle Inequality Theorem using Coordinates

    Homework Statement Prove the Triangle Inequality Theorum using the coordinate system. Homework Equations The corners of the triangles will be at (x1,y1), (x2, y2), (x3,y3) The Attempt at a Solution The proof that I know is proving that |x+y|<=|x|+|y|: -|x|<x<|x|, and...
  9. N

    Finding the minimum area of a triangle

    So i have the points A=(1,0,1) and B=(1,-1,0) and the third corner lies on the line L:(x,y,z) = (t,t,t) and i need to find a triangle with the minimum area possible. My initial approach was to calculate the vector BA and BC and then apply both vectors in the area triangle formula...
  10. J

    Finding angle in triangle and semicircle

    Dear all, right now, I am doing my IGCSE and this question is in my coursebook. I can't think of any way to calculate it... It's part of the section circumference and area of circles. So must be some how solved using this. The semi-circle and the isosceles triangle have the same base AB and...
  11. C

    Right triangle with four jointed links

    Homework Statement A right triangle ABCD is formed from four jointed links. The joint D at the midpoint of the hypotenuse is pushed until it lies on the leg AB. Homework Equations Find the height x when a = 50 mm The Attempt at a Solution give me a hint please cause I'm lost :mad:
  12. C

    Metric Spaces, Triangle Inequality

    I have the following question on metric spaces Let (X,d) be a metric space and x1,x2,...,xn ∈ X. Show that d(x1, xn) ≤ d(x1, x2) + · · · + d(xn−1, xn2 ), and d(x1, x3) ≥ |d(x1, x2) − d(x2, x3)|. So the first part is simply a statement of the triangle inequality. However, the metric...
  13. C

    Circumcentre of an equilateral triangle

    [b]1. Homework Statement [ what is the distance from one corner of an equilateral triangle of sides a to the circumcentre? I can figure out the length from one corner to the opposite side to be sqrt3*a/2 but that's about it. I just can't see how to do this. thanks
  14. L

    Bringing 3 Charges from infinity into a triangle

    Homework Statement How much work is needed to arrange three charges, Q, into an equilateral triangle? The particles are initially infinitely far apart. Take 'a' to be the length of each side of the triangle. Homework Equations U = {(kqq)/L} The Attempt at a Solution I was under...
  15. A

    Proving a 2:1 ratio triangle having a Zero Vector

    Homework Statement I need help once again with this question below: Thanks, aeromat The Attempt at a Solution This is what I have so far: GP = GQ + QP GQ = GP + PQ GR = GP + PR GP + GQ + GR = GQ + QP[-] + PQ[+] + GP + PR + GP GP + GQ + GR = GQ + 2GP + PR I am stuck...
  16. Demon117

    Different orientation on this triangle

    I am trying to get a different orientation on this triangle; specifically I want this output: ********** ********* ******** ******* ****** ***** **** *** ** * Here is the code that I have #include <iostream> using namespace std; int main () { int lines=10; int start =...
  17. G

    Triangle Approximation Derivation

    Homework Statement Here is a drawing with all the needed variables: http://i.imgur.com/192GI.jpgHomework Equations The Attempt at a Solution I have been trying to figure out how this approximation is derived for some time now and have no progress to show for it. Any help in figuring out the...
  18. M

    Applied Trigonometry, Deduce Length of Segment in new triangle.

    Homework Statement When the sun is 74° above the horizontal a vertical flag pole casts an 8.5m shadow on the horizontal. Find the shadow cast when the sun lowers to 62° above the horizontal. Homework Equations The Attempt at a Solution I drew the following diagram...
  19. M

    How can the Newton-Raphson method be used to solve a curved triangle equation?

    I'm having a hard time figuring out what to do here and it feels as though it involves converting the function into polar coordinates but I'm not a hundred percent sure. Essentially, I have a triangle in which two sides are straight and one side is a curved function. Knowing only the length...
  20. A

    Centre of mass of of an isosceles Triangle

    Homework Statement Hi I am really stuck on this problem. It reads as such: A uniform triangular lamina is isocceles and has the line y= 4 as its axis of symmetry. One of the vertices of the triangle is the point (2,1). Given that the x-co-ordinate of the centre of mass of the lamina is -3...
  21. Z

    Find 'k' in Circle Inscribed in a Triangle

    Homework Statement The circle x2 + y2 - 4x - 4y + 4 = 0 is inscribed in a triangle, which has two of its sides along the coordinate axes. If the locus of the circumcentre is of the form x + y - xy + k(x2 + y2)1/2= 0. Find k.The Attempt at a Solution The centre of the given circle is (2,2) and...
  22. G

    Elementary Analysis, Triangle Inequality Help

    Homework Statement Prove that ||a|-|b||\leq |a-b| for all a,b in the reals Homework Equations I know we have to use the triangle inequality, which states: |a+b|\leq |a|+|b|. Also, we proved in another problem that |b|\leq a iff -a\leqb\leqa The Attempt at a Solution Using the...
  23. K

    Pascal's Triangle: 40 Paths to Spell BINOMIAL

    Homework Statement How many different paths will spell the word BINOMIAL in the following arrangement(moving diagonally downwards to the left or right)? ...B ...I I ..N N N O O O O .M M M ...I I ...A ...L L Homework Equations The Attempt at a Solution Starting from B using Pascal's Triangle...
  24. A

    Area moment of inertia of a triangle

    Homework Statement What is the moment of inertia WRT to the x-axis (Ix) of a right triangle with base b and height h where the legs of the triangle coincide with the x and y axis? It would be helpful to know both the single integral and double integral method. Homework Equations Ix=int(y^2 dA)...
  25. D

    The force on one face of a right triangle is ?

    Homework Statement Problem: Find the force on one face of a right triangle of sides 4 meters and altitude of 3 meters. The altitude is submerged vertically with the 4 meter side in the surface. The selections by the way are a. 58.86 kNewton b. 53.22 kNewton c. 62.64 kNewton d...
  26. A

    Center of mass of right triangle, without calculus

    Homework Statement How can I find the center of mass of a right triangle, of height h, length a, and mass m, without using calculus? Homework Equations The Attempt at a Solution I just need a hint on how to start please
  27. M

    Line Integral Around Triangle: Curl or Not?

    Homework Statement Without parameterizing the path, determine what the value of the line integral (integral of F dot dr) is, if C is the closed, oriented path that travels around the triangle with vertices (0,0) (5,2), and (-3,6) and F=yi + xj Homework Equations Curl possiblY? The...
  28. T

    Triangle Geometry: Prove AO/AD+BO/BE+CO/CF=2

    Homework Statement O is any point inside the triangle ABC. AO, BO, CO are joined and produced to meet the opposite sides BC, CA, and AB at D,E and F respectively. Prove that AO/AD+BO/BE+CO/CF=2 Homework Equations The Attempt at a Solution Obviously, O is meant to be the...
  29. T

    Does Point M Lie on Line BC in This Triangle Bisector Problem?

    Homework Statement Let ABC be a triangle in which angle A = 600. Let BE and CF be the bisectors of the angles B and C, with E on AC and F on AB. Let M be the reflection of A in the line EF. Prove that M lies on BC. http://img135.imageshack.us/img135/606/sumu.png The Attempt at a Solution I...
  30. A

    Balancing a circle on a triangle?

    Homework Statement You're given a hoop with mass m and radius R balanced on top of a knife blade. (The diagram looks like a triangle with a circle balanced on the tip.) Find the period of small oscillations. (Yes, that is all the problem says.) Homework Equations Moment of inertia of a...
  31. R

    Optimization problem, triangle

    Homework Statement a line passes through the point (1,1/8) and intersects the positive x-axis at the point A and the positive y-axis at the point B. What is the shortest possible distance between A and B? Homework Equations i came up with three slopes for this line m1=-b/a...
  32. U

    Maximising the area of a triangle of known perimeter

    How would I find the maximum area of a triangle given a fixed perimeter? I assume that it would be an equilateral triangle, but I'm finding it very hard proving it. I started by drawing a triangle of side lengths a, b and P -(a+b) with angles of alpha, beta and gamma. I then used the...
  33. H

    When does equality occur in the inequality (a^2+b^2)cos(α-β)<=2ab?

    Prove that in any triangle ABC with a sharp angle at the peak C apply inequality:(a^2+b^2)cos(α-β)<=2ab Determine when equality occurs. I tried to solve this problem... I proved that (a^2+b^2+c^2)^2/3 >= (4S(ABC))^2, S(ABC) - area but I don't know prove that (a^2+b^2)cos(α-β)<=2ab :(...
  34. A

    Resultant vector of an isosceles triangle?

    Homework Statement what is the resultant vector of an isosceles triangle? Homework Equations R^2=a^2+b^2-4abcos(theta) The Attempt at a Solution The books answer R=2acos(theta/2) Using the formula above, and knowing that a=b in a isosceles triangle I am getting...
  35. P

    Triangle Side Lengths: Find the Third Side with 10 and 14 Units & 30° Angle

    A triangle has sides of length 10 and 14 units respectively, and one of the non-included angles is 30 degrees. Find the possible lengths of the third side. The answers is the product of these lengths.
  36. P

    Triangle Inequality, Integrals

    Is it true in general that: |\int f(x)dx| < \int |f(x)|dx Not sure if "Triangle Inequality" is the right word for that, but that seems to be what's involved.
  37. Z

    Deriving the Distance Between Orthocentre & Circumcentre of a Triangle

    The problem Can anyone explain me a step given in my book? This is actually the derivation of the distance between orthocentre and circumcentre of a triangle using vectors. ABC is a triangle. S is the circumcentre and H is the orthocentre. \vec{SA} + \vec{SB} + \vec{SC} = \vec{SA} +...
  38. J

    How Does Mass Distribution Affect the Center of Mass in a Triangle?

    Homework Statement I know traditionally the center of mass of a triangle is 2/3 the way down its height, but i believe it varies here due to the uneven mass distribution. Here i have to find the center of mass distance from Q http://img508.imageshack.us/f/scn0001p.jpg/ Here is an image...
  39. Z

    Moment of Inertia - equilateral triangle

    Homework Statement [PLAIN]http://img59.imageshack.us/img59/9484/fp5.gif to find the moment of inertia through point A Homework Equations I = \int{r^2dm}The Attempt at a Solution Used a double integral from point A: \displaystyle\int_{0}^{h}\displaystyle\int_{-L/2}^{L/2}\rho(x^2+y^2)dxdy =...
  40. M

    What are the Tensions and Accelerations in a Rotating Triangle Mass System?

    Homework Statement Two point particles of mass m are attached to two vertices of an equilateral triangle made of three massless rigid rods of length l. The third vertex is pivoted to a fxed point, so that the triangle can swing in a vertical plane under the effect of gravity. The system...
  41. D

    CM of triangle with integration

    Homework Statement I need to calculate center of mass of this isosceles triangle using integration: the density is constant. Homework Equations Rcm = integral(xdm) / integral(dm) The Attempt at a Solution I know how to begin: dm = density * dx * dy; x = double-integral(x*dx*dy)...
  42. S

    Masses in a Triangle - Gravitational Force

    Homework Statement Three uniform spheres are located at the corners of an equilateral triangle. Each side of the triangle has a length of 1.23 m. Two of the spheres have a mass of 2.84 kg each. The third sphere (mass unknown) is released from rest. Considering only the gravitational forces...
  43. R

    The Tetraktysal Kissing Triangle (TK_n) & Lower Bounds of Kissing Numbers to D=10

    The "Tetraktysal Kissing Triangle" (TK_n) & Lower Bounds of Kissing Numbers to D=10 The Tetraktysal Kissing Triangle (TK_n) Nickname: "The TetraKiss Triangle" A Fibonacci, Lucas, Tetrahedral Convolution Construction for Lower Bounds of Sphere Packings to Dimension 10 Based Upon the Pythagorean...
  44. G

    Related Rates of a Triangle with Fixed Side Lengths

    Homework Statement Two sides of a triangle of fixed length measure 12m and 15m. The angle between these two sides increases at a rate of 2 degrees per minute. When the angle between these two sides is 60 degrees, at what rate is the third side increasing? Homework Equations That's the...
  45. F

    Triangle Inequality and Pseudometric

    Homework Statement d(x,y)=(a|x_1-y_1|^2+b|x_1-y_1||x_2-y_2|+c|x_2-y_2|^2)^{1/2} where a>0, b>0, c>0 and 4ac-b^2<0 Show whether d(x,y) exhibits Triangle inequality? Homework Equations (M4) d(x,y) \leq d(x,z)+d(z,y) (for all x,y and z in X) The Attempt at a Solution I...
  46. F

    Finding the displacement of a non right triangle?

    Homework Statement A small map shows Atlanta to be 730 miles in a direction of 5 degrees north of east from Dallas. The same map shows that Chicago is 560 miles in a direction of 21 degrees west of north from Atlanta. Assume a flat Eath and use this information to find the displacement from...
  47. P

    Where Is the Center of Mass in a Right Triangle?

    Homework Statement In the figure, a 60 cm length of uniform wire, of 60 g mass, is bent into a right triangle. The x- and y-coordinates of the center of mass, in cm, are closest to... I am going to explain the picture in my textbook. The origin is at (0,0). It goes up 10 cm and to the...
  48. I

    Center of mass: mass's on a right triangle system

    A system consists of three particles located at the corners of a right triangle as shown below. Mass 1 is 7.0 kg, mass 2 is 6.4 kg, and mass 3 is 9.9 kg. If the distances a equals 15.8 m and b equals 2.0 m, then what is the center of mass (in m) in the vertical direction? Y center of...
  49. I

    Center of mass: 2 objects in triangle system

    Three objects of different masses are located at the corners of the triangle shown in the picture. If distance a equals 4.21 m, b equals 1.55 m, and c equals 11.93 m, then what is the center of the mass (in m) of the system in the horizontal with respect to the origin? X center of...
  50. Telemachus

    Optimization: maximize a triangle surface

    See if anyone can help me with this: Among all triangles of perimeter equal to P, find the one with the largest area. (Hint: use the formula A=\sqrt[ ]{p(p-x)(p-y)(p-z)} where P=2p, P is the perimeter). So, I have f|_s , I think that must be solved using Lagrange multipliers, at least I don't...
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