# Triangle Definition and 32 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. ### A formula involving the sum of cosines of the angles of a triangle

Problem Statement : The statement appeared on a website where a different problem was being solved. I got stuck at the (first) statement in the solution that I posted above 👆. Here I copy and paste that statement from the website, which I cannot show : Attempt : To save time typing, I write...
2. ### 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##?
3. ### 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...
4. ### I Necessity of absolute value in Cauchy Schwarz inequality

Reading The Theoretical Minimum by Susskind and Friedman. They state the following... $$\left|X\right|=\sqrt {\langle X|X \rangle}\\ \left|Y\right|=\sqrt {\langle Y|Y \rangle}\\ \left|X+Y\right|=\sqrt {\left({\left<X\right|+\left<Y\right|}\right)\left({\left|X\right>+\left|Y\right>}\right)}$$...
5. ### 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!
6. ### Triangle and hyperbola

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...
7. ### B Triangle of powers

Recently I saw this clip on YouTube which has a revolutionary idea of mathematical notations,What do you think about it?
8. ### Find the ratio of two line segments in a triangle

1. The problem statement, all variables, and given/known data Triangle ABC has a point D on the line segment AB which cuts the segment in ratio AD : DB = 2 : 1. Another point E is on the line segment BC, cutting it in ratio BE : EC = 1 : 4. Point F is the intersection of the line segments AE and...
9. ### I What is the maximum triangle?

I know that if a triangle have it edge // to each orther then it í the maximum triangle. Pls explain i don't understand hơ thí even possible
10. ### How to find the angles of a triangle in a semicircle?

Homework Statement Homework Equations d(y)/d(x) --> max area area of triangle = 1/2 . base . height The Attempt at a Solution for number (2) [/B] x^2 + y^2 = r^2 --> circle equation base = 2R, height = y Area = 1/2 . 2R . y area = 1/2 . 4. √ (r^2 - x^2) area now is half of max = 2...
11. ### A circle is circumscribed around triangle ABC, find length?

some formula related I tried to draw the problem can anyone give me clue how to solve it?
12. ### Finding an angle of a triangle as function of another angle

1. Find α(β) given that the sum of the 2 sides= ##(x+y)## and its third, ##z## is a constant for 0<β<180. You can imagine that there's two pieces of string connected between two points. One string is as long as the distance between the two points while the other string is longer. If you...
13. ### Area of Hexagon - Geometry Challenge

Determine the area of the painted hexagon, knowing that the area of triangle ABC is 120cm^2 IMG Link: https://m.imgur.com/a/WtdsW I tried using Heron´s formula, but just ended up with a bunch of terms and one more variable. Sidenote: I guess part of it is figuring out that the side lenghts...
14. ### Limits of Integration of a Triangle

Homework Statement Suppose you have a Triangle with the vertices, (0,0) (1,1) and (0,1). Integrating along that path. I have some differential function dZ where Z = Z(x,y) Homework Equations The Attempt at a Solution [/B] If I need to integrate, then I need to find the limits of...
15. ### Circle inscribed in a triangle exercise

Homework Statement In the drawing you can see a circumference inscribed in the triangle ABC (See the picture in the following link). Calculate the value of X https://goo.gl/photos/CAacV2dJbUrywfXv9 2. The attempt at a solution It seems I found a solution for this exercise with the help of...
16. ### Comp Sci Fortran: Find a triangle's side, plus angles

The statement: In every ABC triangle, the laws of sine (a/sinA = b/sinB = c/sinC) & cosine (c2 = a2 + b2 -2*a*b*cosC) are valid, where a, b & c are the sides opposite to A, B & C respectively. Write a program that calculates and prints on the computer's screen, the length of c, and also the...
17. ### 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...
18. ### 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 Equations The Attempt at a Solution I know that the magnitude of the cross product of any two vectors...
19. ### 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. ### 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...
21. ### 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...
22. ### 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...
23. ### Area of a triangle under a curve

Homework Statement The tangent line of a curve y=e^(-x) intercepts the axises at points A and B. What is the maximum area of a triangle AOB considering O as the origin. Homework Equations Ar= xy/2 The Attempt at a Solution Derivative of this function is y'=-e^(-x) I took the formula of the...
24. ### Static forces And equilibrium

1. Homework Statement Find forces P,F,and T Homework Equations Fx = 0 Fy = 0 The Attempt at a Solution So far I only got force T from getting the moment from point b. Don't know how-to start getting force P and F since they're both at the same point. Any help will be much obliged thank...
25. ### [Euclidean Geometry] Kiselev's Plainimetry Question 242

Homework Statement Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points D and Ε with MA and MB respectively. Prove that (1) the perimeter of ▲DME, and (2)...
26. ### Side of a triangle (geometry)

Homework Statement As shown in the diagram below, the shape consists of a square and a circle with centre Q. Given that QM = 3 cm, prove that MN = 6 cm. Known data: -- triangles APB and BQC are congruent -- angle BMC = 90 -- triangles BMQ and BNA are similar and right-angled 2. Homework...
27. ### Given isosceles triangle, find sin (A-C)

Homework Statement Triangle ABC have side AB = 10 cm, AC=BC = 13cm, so sin (A-C) is... 2. Homework Equations sin (A-C) = sin A cos C - cos A sin C The Attempt at a Solution I see that the triangle can be split into two right-angle triangles. But, sin (A-C) ?? How to get that?[/B]
28. ### Getting a triangle from trigonometric function

How can I get a right triangle from the inputs and outputs of trigonometric functions? For example: sin(x) = y The triangle would have one angle as x and the opposite edge of the triangle would be y/hyp etc. How can I get all of these values from any trigonometric function? Please tell me if I...
29. ### Exact value

The problem A right triangle has an angle a and we know that ##cos \ a = \frac{1}{3}##. What is ## tan \ (90°-a) ## The attempt I know that the ration between the adjacent side and the hypothenuse is 1/3. I am not interested in the real lengths of the sides. I can therefore calculate the...
30. ### Area of this triangle?

This is supposed to be a simple question. However, I forgot a lot of the basics and rules I have to follow. I tried to workout the height based on the area: 0.5 x 3 x h = 30 h = 20 But couldn't figure out the rest. Then I thought about going by ratio (not from knowledge but out of...