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. R

    Create a Triangle with Asterisks in C: Step-by-Step Guide

    < Mentor Note -- code tags have been added for better readability. In the future, please use code tags. Thank you. > 1. Homework Statement #include <stdio.h> void escreve_linha(int n) { int i; for( i = 0 ; i < n ; i++ ) { printf("*"); } printf("\n"); } void...
  2. M

    I Fnd the area A of the triangle with the given the vertices

    (0, 0), (3, 5), (1, 8) Find the slopes and equations for each line (0,0) ----> (3,5) = 5/3x (0,0)---->(1,8) = 8x (1,8)---->(3,5) = -3/2x+ 19 Then I set up the integrals (on x) Integral sign from 0 to 1 (8x-5/3x)dx + Integral sign from 1 to 3 [(-3/2x+19)-5/3x) dx I got 117/4 as an...
  3. M

    MHB Draw Sketch: Right Triangle Area = Half Parallelogram Area (Party)

    Any thoughts on the sketch? (Party) Many Thanks :)
  4. L

    What are the lengths of lines M to M1 and M to M2 in this geometry problem?

    Homework Statement line m2-to-m is 3km longer than line m1-to-m what are lengths for ##M~~M_2## and ##M~~M_1## Homework Equations pythagorean theorem hopefully can be used The Attempt at a Solution use the picture to your advantage in hopefully creating a valid system of equations. With...
  5. C

    Related rates - finding hypotenuse of triangle

    Homework Statement A plane flying horizontally at an altitude of 3 mi and a speed of 480 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 4 mi away from the station. (Round your answer to the nearest whole...
  6. M

    MHB Tackling a Difficult Triangle Exercise

    I am having a very difficult time comprehending this exercise and any direction would be appreciated. Feel completely lost! Not necessarily asking for the answer just want to know how to tackle this. This program will output a right triangle based on user specified height triangleHeight and...
  7. Nipuna Weerasekara

    Can a triangle be formed with these length constraints?

    Homework Statement There is a triangle with sides $$ 3,3r,3r^2 $$ such that 'r' is a real number strictly greater than the Golden Ratio. Is this statement true or false...? Homework Equations $$Golden \space Ratio = \phi = 1.618... $$ The Attempt at a Solution Actually I have no clue at all...
  8. R

    I Primality test from pascal triangle

    In pascal triangle, if all the elements of a row(except 1 in both end) are divisible by the row number, then the row number is a prime. Or if the coefficient of binomial expansion (except 1's) are divisible by the power, then the power is a prime. It is inefficient to check all the coefficient...
  9. M

    MHB Proving angles are equal in a triangle in a circle.

    hi can someone help me work this out. i think it has something to do with the exterior angle of a triangle is equal to the sum of the interior angles but i can't work it.
  10. T

    MHB Find unknown angles of a triangle

    Dear friends, I am unable to find out the unknown angles for the following triangle which I attached with this post. Angle BAD and angle BCD are the unknown angles need to be calculated. Given that lines AB=BC=CD and angle CDE = 108 degrees From my calculations: angle ADC = 180 - 108 = 72...
  11. D

    Calculate the electric field using superposition

    Homework Statement Find the expression for the electric field at point M(a,a,0) if the linear charge density is known ( ##Q'## ) Homework Equations 3. The Attempt at a Solution [/B] I tried something like this and would like your feedback on it. I separated the triangle into three parts...
  12. Kernul

    Build right triangle with two points and a line

    Homework Statement Given the points ##A (1, -1, 0)## and ##B (4, 0, 6)##, find the point ##P## of the line ##s## so that the triangle ##ABP## is a right triangle in ##B##. Calculate the area of the triangle. ##s : \begin{cases} x = 1 + 4t \\ y = 2 - 3t \\ z = 3 \end{cases}## ##\vec v_s = (4...
  13. M

    MHB Prove that triangle BAD is isosceles and....

    Problem In the $\triangle ADC$ , $\angle DAC$ or angle $A$ is a right angle, E is the midpoint of AC . The perpendicular drawn to $AC$ from $E$ meets $DC$ at $B$ i.Drawn the given information in a figure & prove that $\triangle BAD$ is isosceles ii. $AC^2+AD^2=4AB^2$ Diagram Where do...
  14. M

    MHB Ratio of the area of triangle in terms of another triangle

    :D I have trouble in determining the ratio of the area of $\triangle PST$ in terms of $\triangle PQR$ In the triangle PQR $QT=TR$, $PS=1 cm$ , $SQ=2 cm$ , How should I be writing the area of $\triangle PST$ in terms of $\triangle PQR $ What is known by me : Since...
  15. Gjmdp

    Finding the Radius of a Tangent Circumference in a Right Triangle

    Homework Statement Let AC=5 and BC=12. In the triangle ABC, with angle C=90, point M is in AC. A circumference with center M and radius r is tangent to AB and tangent to BC in C. Set r. Homework Equations This should envolve basic trigonometry, and Thales' theorem; but not sure ( if I knew the...
  16. anemone

    MHB Right-Angled Triangle Inequality

    Show that if $a,\,b$ and $c$ are the lengths of the sides of a right triangle with hypotenuse $c$, then \frac{(c − a)(c − b)}{(c + a)(c + b)}\le 17 − 12\sqrt{2}
  17. math4everyone

    Electric potential at the vertex of a triangle

    I am stuck with this problem: The right triangle shown with vertex P at the origin has base b, altitude a, and uniform density of surface charge σ. Determine the potential at the vertex P. First find the contribution of the vertical strip of width dx at x. Show that the potential at P can be...
  18. steele1

    Prove area of triangle is given by cross products of the vertex vectors....

    Homework Statement The three vectors A, B, and C point from the origin O to the three corners of a triangle. Show that the area of the triangle is given by 1/2|(BxC)+(CxA)+(AxB)|. Homework EquationsThe Attempt at a Solution I know that the magnitude of the cross product of any two vectors...
  19. S

    Prove this is a right triangle in a sphere

    Homework Statement Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle Homework Equations |AB|^2 = |AP|^2 + |PB|^2 |AB}^2 = 4r^2 The Attempt at a Solution Not sure if showing the above equations are true is the...
  20. M

    MHB Maximizing the area of a triangle

    can some1 show me how to do this the answer to the first part is √x²+√16/x²
  21. M

    MHB Find the radius of the sector adjoining a triangle

    https://i.imgsafe.org/5172eee94d.jpg For a closer look click https://i.imgsafe.org/5172eee94d.jpg We know that the area of the sector should be $\frac{40}{360}$*$\frac{22}{7}$*$r$*r Any ideas on how to begin? Many Thanks:)
  22. M

    MHB Construct vertex D of an acute angled triangle

    For a closer look click here Any Ideas on how to begin? Many Thanks :)
  23. F

    Isosceles triangle with angle bisector

    Homework Statement Find x[/B] Homework Equations a+b+c=180[/B]The Attempt at a Solution ADB= 65+65+50=180 How do I find x in DBC?
  24. F

    What is the measure of an isosceles triangle angle when given its base angles?

    Homework Statement Find x[/B] Homework Equations a+b+c=180 [/B]The Attempt at a Solution So RP=RQ, the base angles are x and (146-x), normally the base angles are the same but in this scenario they are different. x+146-x+?=180
  25. F

    What are the sizes of angles in a scalene triangle with given expressions?

    Homework Statement The three angles of a scalene triangle are x degrees, (x-12) degrees and (2x+6) degrees. What sizes are the angles?[/B] Homework Equations Triangle = a+b+c =180 Scalene = no equal size or angles[/B]The Attempt at a Solution So what do you do with (x-12) degrees and (2x+6)...
  26. Mr Davis 97

    B Proof that exterior angles of a triangle sum to 360

    So I am working on this simple proof, but am confused about the term "external angle." The problem says that if ##a##, ##b##, and ##c## are external angles to a triangle, then ##a + b + c = 360##. However, is seems that the vertex of each triangle has two possible external angles, since there...
  27. M

    MHB Right triangle, feet of altitude, angle bisector and median

    Let $\triangle ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$, and $AB < AC$. Let points $D, E, F$ be located on side $BC$ such that $AD$ is the altitude, $AE$ is the internal angle bisector, and $AF$ is the median. Prove that $3AD + AF > 4AE$ My solution. Can you check it is...
  28. RoboNerd

    Question about determining the angles of triangle given two vectors

    <<Mentor note: Missing template due to originally being posted elsewhere>> Hello everyone. I have the following problem: Determine the angles of a triangle where two sides of a triangle are formed by the vectors A = 3i -4j -k and B=4i -j + 3k I thought that I would find the third side being...
  29. F

    MHB Proof Quest: Non-Equilateral Triangle Enclosing Point Z

    I am struggling with this question, it would be easy enough if the triangle was equilateral but that is not necessarily the case. Let (ha, hb, hc) be heights in the triangle ABC, and let Z be a point inside the triangle. Further to this, consider the points P, Q, R on the sides AB, BC and AC...
  30. F

    MHB Triangle side terms and area inequality

    Currently revising for my A-Level maths (UK), there is unfortunately no key in the book; Given the triangle with sides a,b,c respectively and the area S, show that ab+bc+ca => 4*sqrt(3)*S I have tried using the Ravi transformation without luck, any takers?
  31. J

    B Proofs: Hypotenuse is the longest side of a right triangle

    I want to prove that the hypotenuse is the longest side of a right angled triangle. Could people check that the proof I'm giving is correct? Say the hypotenuse is of length ##c## and the other two sides are of length ##a## and ##b##. First of all, we obviously have: ##a^2 + b^2 > a^2 \quad##...
  32. C

    Understanding Mohr's Circle Formula: Angle Placement Outside the Triangle

    Homework Statement this is actually mohr's circle formula, forget about the theory,let's focus on the mathematics part. I couldn't understand why the tan( 2 θs1) = -(σx -σy) / 2τxy ? 2θs1 is outside the triangle Homework EquationsThe Attempt at a Solution theorically, tan(180-α )= -tan( α ) ...
  33. Virang807

    I Question about Hydrostatic Force?

    Hello! I am currently in Calculus 2 and we are dealing with hydrostatic force. I get the integration that happens but I always seem to have trouble with solving the word problems. With this problem, I realize that I use integrate by making an infinite number of rectangles on the 2 triangles. I...
  34. anemone

    MHB Inequality Involves The Sides Of Triangle

    Let $a,\,b$ and $c$ be the sides of a triangle and $x,\,y$ and $z$ are real numbers such that $x+ y+ z = 0$. Prove that $a^2yz +b^2xz+c^2xy\le 0$.
  35. Z

    Portion of Altitude of a Triangle Inscribed in a Circle

    Homework Statement In the figure Q image2.jpeg (attached), equilateral triangle ABC is inscribed in circle O, whose radius is 4. Altitude BD is extended until it intersects the circle at E. What is the length of DE? Solution figure is attached. They formed a right angled triangle & calling it...
  36. M

    MHB Derive a Generalised Formula for c-d with Respect to θ

    In the above given triangle when θ = 0 then b=c But when θ = 90 then d=0 Since d is the projection of b How we can derive a generalised formula for d or c-d with respect to θ Plese may kindly be elaborated
  37. alaa amed

    Average rate of change of the area of the triangle?

    Homework Statement An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t)=(cos(t),sin(t)) . Assume 0 < t < π/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle...
  38. W

    MHB Can you solve this right triangle puzzle with a unique angle ratio?

    B b-a D a c-b+a C b A Right triangle ABC, with the standard a, b, c side lengths. Angle BAC = 2u degrees. Point D is on the hypotenuse AB, such that: BD = b-a, angle BCD = 3u degrees. Calculate u.
  39. L

    A Why Is the Discriminant Non-Positive in the Triangle Inequality Proof?

    In the derivation of triangle inequality |(x,y)| \leq ||x|| ||y|| one use some ##z=x-ty## where ##t## is real number. And then from ##(z,z) \geq 0## one gets quadratic inequality ||x||^2+||y||^2t^2-2tRe(x,y) \geq 0 And from here they said that discriminant of quadratic equation D=4(Re(x,y))^2-4...
  40. BiGyElLoWhAt

    Exploring the Benefits of Ionocrafts and their Equilateral Triangle Shapes

    Hmmm maybe this goes here, maybe it doesn't. https://en.wikipedia.org/wiki/Ionocraft Here is a picture that depicts basically what I'm talking about. From what I understand, Ionocrafts work on N3L, and shoot ions down, pushing the craft up. This makes me think that the more charges you have...
  41. H

    Sum of the angles of a spherical triangle

    Homework Statement What is the sum of the angles of a spherical triangle formed on the surface of a sphere of radius R? The triangle is formed by the intersections of the arcs of great circles. Let A be the area of the surface of the sphere enclosed by the triangle. This question is a...
  42. P

    Charged particle in centre of equilateral triangle

    Homework Statement 3 charged particles located at equilateral triangles corners has charge ##q_1=q_2=q_2=4*10^{-6} C ## The 4th particle with charge ## q_4=3*10^{-4}## is placed at bisectors crossing. The distance between q1 and q4 is 3.46 cm .What is the net force on q4? Homework Equations...
  43. 1

    Why is this triangle not right? (sine -1 is bigger than 1)

    the black triangle is given, and red triangle is me trying to work it out why is my trangle wrong? and what is the right triangle? someone is sliding down a slope, with 2ms-2 of acceleration (not actually given which direction, maybe I'm wrong there, but the person is acelerating at 2ms-2 down...
  44. anemone

    MHB Prove Equilateral Triangle from Cosine Equality

    Prove that if in a triangle $ABC$ we have the following equality that holds $2\cos A \cos B \cos C + \cos A \cos B + \cos B \cos C + \cos C \cos A = 1$ then the triangle will be an equilateral triangle.
  45. S

    Moment Question / inclined triangle

    Homework Statement In this I tried to resolve the components. So first thing, I converted the 12kg into Newtons so it would be 117.72 Newtons. Then found the perpendicular distance which is to g: cos(30)x1 then multiply the answer by 117.2N to give the weight down as 102N. As the moment...
  46. M

    Parallel transportation of a vector along a closed triangle

    Hello everyone, I am trying to solve exercise 7.21 in the "Hobson, Efstathiou, Lasenby, General Relativity. An introduction for physicists." What is asked is to show that the parallel transportation of a vector, along a closed triangle on a 2-sphere, results in an vector orthogonal to the...
  47. prashant singh

    B Can we say that a line is a particular case of a triangle

    I think we can say that a straight line is a triangle in which sum of two side equals third side in which two angle equals 0° and the middle angle equals 180°, but can we say that a striaght line is a triangle with two angle equals 90°, I am saying this because my sister is not beliving that...
  48. Titan97

    Geometry Book for learning properties of triangle and circle

    Can you suggest a book that discusses properties of triangles and circles? (Like properties and theorems on circumcircle, excircle, nine point circle, etc). Most of the geometry books are either to basic or too advanced. I have read a book on complex numbers by Liang Shin Hahn. But the...
  49. V

    How Do You Solve the Ambiguous Triangle Case in Trigonometry?

    Homework Statement kettletown is 27km from teatown and 32km from coffee town. The angle from kettletown to coffee town to teatown is 29o. Determine the two possible distances between teatown and coffee town. Homework EquationsThe Attempt at a Solution The only problem I'm having is trying to...
  50. e2m2a

    I Right Triangle and the Pythagorean Triple Formula

    Given: A right triangle and all the sides of the triangle are whole numbers. Does this imply that all the sides of the triangle can only be found by using the Pythagorean triple formula? Another words, is it possible that a right triangle can exist with whole number sides that escape the...
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