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. M

    MHB Find the scale factor of triangle ABC to triangle DEF

    Triangles ABC and DEF are similar. Triangle ABC has a perimeter of 16cm. Triangle DEF has side of 6cm, 8cm and 10cm. What is the scale factor of triangle ABC to triangle DEF? A. 1/2 B. 1/3 C. 2/3 D. 3/2 E. 2/1 I concluded the answer is D. Am I correct?
  2. Monoxdifly

    MHB How Do I Find the Area of a Triangle with Non-Intersecting Vertices?

    What's the area of the triangle? It's hard because the vertices aren't in the intersections of horizontal and vertical lines, so I have a hard time determining the side lengths, and it's also for Elementary Students Math Olympiads too.
  3. L

    B Calculating Triangle Area in Relativity Theory

    Area of triangle from picture https://en.wikipedia.org/wiki/Special_right_triangle#/media/File:45-45-triangle.svg is ##A_0=\frac{1}{2}##. If that triangle staying still in system S' and S' moving across one of the sides of length ##1## in respect to system ##S## area of the triangle in the...
  4. S

    How to Find the Ratio of Areas in a Scalene Triangle?

    I do not know where to start. I draw scalene triangle and assign each coordinates to the vertices. Tried writing something but none working. Please give me a hint to start. Thanks
  5. Monoxdifly

    MHB Find Length of Side in Right Triangle w/ 45°, 90°, and 45° Angle

    The triangle ABC is a right triangle with A as the right angle and BD is the bisector of angle B. If AB = 12 cm and BC = 15 cm, the length of AD is ... A. 3 cm B. 4 cm C. 5 cm D. 6 cm It was a question for a 9th grader and the book hasn't covered trigonometry by name yet (As in, they don't know...
  6. G

    MHB Analytic geometry proof with triangle.

    Point D divides side AC, of triangle ABC, so that |AD|: |DC| = 1:2. Prove that vectors \vec{BD} = 2/3 \vec{BA} + 1/3 \vec{BC}.
  7. Arman777

    I Deriving the area of a spherical triangle from the metric

    The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$ Is there an equation to describe the area of an triangle by using metric. Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.
  8. A

    I Help finding a triangle inside a triangle

    I would like to find the triangle ##\bigtriangleup DEF## in the plot below that is inside the right triangle ##\bigtriangleup ABC## given ##\overline{AB}=3, \overline{AC}=4## with ##\overline{BD}=\overline{DE},\overline{AE}=\overline{EF}, \overline{FC}=2\overline{DF}##. However, I'm finding it...
  9. D

    I Geometry question with a triangle

    AB=AC. P is on ac such that AP=3PC. Q on CB such that CQ=3BQ. Need to find the length of PQ. I know i can use the Cosine theorem, but the answer is without Cosine.
  10. B

    MHB Trigonometry: Right-angled triangle

    Hi Guys I need help with this question so if you can help me with the answer please please.
  11. Z

    MHB Largest Quadrilateral inscribed in Scalene Triangle

    I have a scalene triangle: A: 75.04 B: 66.9 C: 41.13 The first thing I need to do is move just lines A and C in towards each other .5 and recalculate all sides. Then I need to inscribe the largest quadrilateral that will fit while having one side being no shorter than 7.5, with the entire...
  12. M

    A Triangle homography is possible I am a first?

    I solve linear equation for getting rotation and position of triangle without rotation and translation matrix. And try it. Just need a AB, BC, AC lenght. Somebody do this before?This is demonstration with camera canon eos 600d. But I don't solve lens distortion yet.
  13. akashpandey

    I Triangle law of vector addition and the Pythagoras theorem

    i know its pretty basic but please give some insight for triangle law of vector addition and pythgoras theorem. becuase ofcourse if you use traingle law to find resultant it will be different from what is pythagoras theorem
  14. G

    MHB Smallest possible area of triangle

    Hi, I'm trying to work out this question, and the answer I'm coming up with isn't right. Can anyone help me understand the calculation used to work this out?
  15. G

    MHB Solve for the angles in a triangle

    Hi, Could someone check that I'm right with this one, or put me right! I've worked the value out as x=30 and y (2x) = 60. I've come to this as I think it's an isosceles triangle so the base angles would be equal. Am I right? Thank you!
  16. Monoxdifly

    MHB [ASK] Find the ratio of the area of triangle BCH and triangle EHD

    A parallelogram ABCD has angle A = angle C = 45°. Circle K with the center C intercept the parallelogram through B and D. AD is extended so that it intercepts the circle at E and BE intercepts CD at H. The ratio of the area of triangle BCH and triangle EHD is ... Here I got that triangle BCH...
  17. G

    MHB How do I find the angles in a triangle?

    Hi, I'm trying to work out how I'd calculate the values in the below. Rather than just have the answer, I'd really like to understand how I'd calculate this. Thank you in advance!
  18. SamRoss

    I Necessity of absolute value in Cauchy Schwarz inequality

    Reading The Theoretical Minimum by Susskind and Friedman. They state the following... $$\left|X\right|=\sqrt {\langle X|X \rangle}\\ \left|Y\right|=\sqrt {\langle Y|Y \rangle}\\ \left|X+Y\right|=\sqrt {\left({\left<X\right|+\left<Y\right|}\right)\left({\left|X\right>+\left|Y\right>}\right)}$$...
  19. Foppe Hoekstra

    I Accelerated Triangle and Length Contraction

    Two astronauts, Neil and Michael, visit a solid not revolving planet. They mount a jet engine on this planet to get it turning around its axis. Before starting the engine they put three dots on the surface with the help of an isosceles triangle, which measures 1 by 1 meter. Two dots are placed...
  20. kaloyan

    Find this angle given the triangle's Orthocenter

    ##AD## is diameter, thus ##\angle ACD = \angle ABD = 90^\circ##. Also ##HBDC## is a parallelogram because ##HC||BD, HB||CD##. It seems useless and I don't know how to continue. Thank you in advance!
  21. Cheesycheese213

    B What is the Area of a Sierpinski Triangle?

    I was trying to find some sort of pattern in the triangle (below) to graph it or find some equation, and I thought maybe measuring something would be a good idea. I was okay just calculating the area for the first few iterations, but then I got confused on how I was supposed to represent like...
  22. F

    MHB Find area of a triangle in coordinate plane

  23. W

    MHB Find Side Lengths of Pythagorean Triangle ABC with Square DEFG Inscribed

    Pythagorean triangle ABC with area 8214 has square DEFG with sides = 60 inscribed in it. Side DE of the square lies on the hypotenuse. Find the triangle's side lengths. B E D F C G A
  24. S

    MHB Area of Triangle ABC: Find the Solution

    ABC is a triangle. AC=4cm; AB=3cm; A=60 degrees. I need help finding the area of triangle ABC.
  25. J

    Comp Sci Computing the Length & Area of a 3D Triangle

    I have computed the total length of a 3D triangle and its area. The code is shown below. I want to use file output instead of cout. The file name, cw2task1output, was just given as part of the task, in this case should I make an empty text file named cw2task1output then attach it to the resource...
  26. C

    How Do You Calculate the Axes of a Hyperbola Given Its Foci and a Point on It?

    Homework Statement Points E and F are the focuses of the hyperbola and point X are on the hyperbola. Determine the size of the main and minor half-axes of the hyperbola. Homework Equations x2 = e2 - f2 x = 8 The Attempt at a Solution I think that eccentricity is 4 units (x/2). But I don’t...
  27. JD_PM

    Evaluate an integral over a triangle

    Homework Statement Let D be the triangle with vetrices ##( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 )##. Evaluate the integral : $$\iint_D e^{\frac{y-x}{y+x}}$$ Homework EquationsThe Attempt at a Solution [/B] The answer to this problem is known (...
  28. anemone

    MHB Challenge question on equilateral triangle: Prove ∠DBA=42°

    In an equilateral triangle $ABC$, let $D$ be a point inside the triangle such that $\angle BAD=54^\circ$ and $\angle BCD=48^\circ$. Prove that $\angle DBA=42^\circ$.
  29. U

    MHB Right Isosceles triangle: solve for the length of the unknown legs

    For a right isosceles triangle (45-45-90) of hypotenuse 1, solve for the length of the unknown legs. Give an exact answer and rationalize the denominator in the final answer.
  30. ino

    Find the electric field at the point P on a right triangle

    Homework Statement https://www.physicsforums.com/threads/find-the-electric-field-in-the-point-p-of-a-right-triangle.965285/#post-6125768 knowing that the three charges are equal and that the angles of the triangle are 90°, 45°, 45°. Homework Equations The Attempt at a Solution I tried...
  31. K

    Problem when calculating the center of mass of a triangle

    Homework Statement if the black dot is assumed by (0,0).find the center of mass coordinate of this triangle [/B] i'm sorry but since the pic won't show ill attach the link here https://ibb.co/4Ptw5T7 <Moderator's note: picture added> Homework Equations centroid is 2/3 of median [/B] using...
  32. CrosisBH

    Finding an area of a triangle formed by three points

    Homework Statement P(3, 0, 3), Q(−2, 1, 5), R(6, 2, 7) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR Homework Equations A = \frac{1}{2}|\vec{AB}\times\vec{AC}| Source...
  33. R

    MHB Change bisector from one triangle to another

    LB is bisector of triangle ABC. suppose, there is a new triangle that called AED. alpha is a angle between LB to AD that is bisector of AED. The triangles can be on any place in the plane. What the value biggest angle (alpha) can be?
  34. R

    Prove the sums of the angles of any given triangle = 180°

    Homework Statement Prove that if A,B,C, are the angles of an arbitrary triangle, then m(A)+m(B)+m(C) = 180 degrees by the following method: From any vertex draw the perpendicular to the line of the opposite side. Then use the result already known for right triangles Homework EquationsThe...
  35. N

    Finding Angle C in an ABC Triangle

    Homework Statement ABC is a triangle The angle A = 30 degrees AB = 7cm BC = 5 cm C is an acute angle. Find the size of angle C? Homework Equations I need to use the cosine rule here Cos C = (a^2 + b^2 - c^2) / 2ac The Attempt at a Solution The problem is I do not know the size of side b? So...
  36. B

    MHB Calculating Triangle Side Length with Known Angles and Radius

    Hi there. Can someone tell me how to calculate the length of "a", shown in these drawings? "r" is the radius of the corner, so these 2 sides have the same length. "C" is 90 deg angle "B" is known (the angle of the corner) Here are the diagrams. First example 45 degree corner, second example 75...
  37. E

    MHB What is the area of an isosceles triangle with side lengths 6, 6, and 4?

    Side lengths are a=6, b=6, c=4. Find the area A=1/2*b*h I split the triangle in half to find the height. Since the base is 4, that divided the base into 2: 2^2+h^2=6^2 4+h^2=36 h^2=32 h=4*sqrt(2) =========== 1/2*4*4*sqrt(2)=area of 8*sqrt(2) Did I do this correctly? I do not have an...
  38. E

    MHB Triangle ABC has area 25*sqrt(3). if Angle BAC=30 degrees, find |AC|=|BC|=?

    Triangle ABC has area 25*sqrt(3). if Angle BAC=30 degrees, find |AC|=|BC|=? the answer I got was 10*4th root(3) Is this correct? I am asking because someone other than my professor wrote the study guid for us for the final and I am not 100% sure what |AC|=|BC| means as my professor never...
  39. K

    MHB Area of Triangle ABC: Find the Answer Here

    what is the area of triangle ABC in the attached? answer is 18 i can not construct any similar triangles here. all i can see is area of ACD is 3 times area of ABD but how does it help me...
  40. K

    MHB What is the length of x in this triangle?

    Hello, what is the length of x in the attached? I tried the following: of course BE = 5, and also the point D must be the center of internal circle that is tangent to triangle and from there i came up with some equations together with the side ratio formulas of angle bisectors but it didnt...
  41. K

    MHB Find Length X in Triangle: Learn How to Calculate It

    what is the length x? (FH and HC perpendicular too, which I missed to write) I am totally stuck can not make any progress on this question. Answer should be 5. I don't know how to obtain it
  42. K

    MHB What is the minimum value for x in this triangle with angle bisectors AD and BD?

    Hello, In the attached, what is the minimum integer value x can take? AD and BD are angle bisectors the answer is 16 - but i do not know how they did it I am totally stuck, could not think of anything here. The angle bisector formula I know does not fit here
  43. YoungPhysicist

    B Triangle of Powers: Revolutionary Math Notation

    Recently I saw this clip on YouTube which has a revolutionary idea of mathematical notations,What do you think about it?
  44. Zack K

    Centroid of an isoceles triangle

    Homework Statement Where is the center of mass of an isoceles triangle? Homework Equations xcm=∫xdV/V (where V is the volume of the triangle) The Attempt at a Solution The representation of the sides is what I'm confused with. Flipping the triangle to it's side is what's recommended to be...
  45. E

    MHB Law of Cosines: Solve for angles given the sides of triangle

    I've attached the problem and my work. When I enter cos^1(6.890625) I get an error, but 6.9 is also not the answer and Does Not Exist is also not an acceptable answer. So where I am going wrong with this?
  46. Y

    MHB What is the Correct Way to Calculate the Area of a Triangle?

  47. Y

    MHB Show that set of points form right-angled triangle

    I was thinking of using Pythagoras here but it didn't get me far Any suggestions?
  48. Robin04

    Triangle determined by arithmetic and geometric sequences

    Homework Statement Determine the triangles where the sides are consecutive elements of a geometric sequence and the angles are consecutive elements of an arithmetic sequence. Homework Equations The Attempt at a Solution I don't really know how to approach this problem, what the solution would...
  49. Greg

    MHB What is the Image of Point P and Line Segment TJ in Triangle Mapping Problem?

    If triangle JPT ----> triangle J`P`T`, what is the image of P? The image of TJ ? Any help would be awesome. (Posted on behalf of Robbie Marsh.)
  50. R

    MHB Find Angles of Spherical Triangle $\mathcal{P}$

    Consider the spherical triangle $\mathcal{P}$ with vertices $P_1 = (1,0,0)$, $P_2 = (0,1,0)$ and $P_3 = (1/\sqrt{3}, 1/\sqrt{3},1/\sqrt{3})$. Find the angles $\phi_1, \phi_2, \phi_3$ of $\mathcal{P}$ at $P_1, P_2, P_3$ respectively. I know the cosine angles are $\cos(\theta_1) = 0$...
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