Questions tagged [triangles]

For questions about properties and applications of triangles.

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Two possible triangle centers.

For every point P in a triangular region T, consider the shortest and longest chords of T that pass thru P. Lets define 'Chord ratio(P)' = ratio between lengths of longest and shortest chords thru P. ...
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The points A $(0,0)$, B$(\cos(\alpha),\sin(\alpha))$ and C $(\cos(\beta),\sin(\beta))$ are the vertices of a right angled triangle.

$(0,0)$, B$(\cos(\alpha),\sin(\alpha))$ and C $(\cos(\beta),\sin(\beta))$ are the vertices of a right angled triangle. Derive a relation between $\alpha$ and $\beta$." /> I tried using the slope ...
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How can I prove that two lines intersect at a circle?---an extended observation.

I’m referring to this problem: How can I prove that two lines intersect at a circle?? I decided to graph it on GeoGebra. By accident, I did not draw it correctly. The line $RS$ was not the angle ...
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-3 votes
2 answers
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Calculate the length of hypotenuse of a right triangle in the complex plane

As the Pythagorean theorem does not work, Base = 1 Altitude = i Hypoteneuse^2 = 1^2 + i^2 = 0? How can this be calculated?
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Angle dependencies between equilateral and right-angled triangle

Given an equilateral triangle $\triangle ABC$ and a right-angled triangle $\triangle ABD$ where $\angle ADB$ is the right angle and, therefore, the hypotenuse $AB$ is shared with the $\triangle ABC$. ...
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Formula for angles $ \left({\cot}^{-1} \left(\frac{{AD}-{AC}}{{BD}}\right)-{\cot}^{-1} \left(\frac{{AD}}{{BD}}\right)\right)\cdot \frac{180}{\pi}$

If we have triangle $\varDelta ABC$, with base $AC$ and height $BD$,we can calculate the angle $\angle ABC$ expressed in degrees,with the formula $$ \left({\cot}^{-1} \left(\frac{{AD}-{AC}}{{BD}}\...
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The answer key of my McGraw Hill Geometry Workbook says a weird thing [closed]

On question 4 of exercise 2.4 of the McGraw Hill Geometry Workbook, which is "What conclusion can you draw from the statements below: If a triangle is isosceles, then it has two congruent sides. ...
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1 answer
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Formalising a Geometric Idea Using Limits

If we choose two points on a circle, they form an arc and a chord. We know that if the angle subtended at the centre of the circle by the two points is small, the arc length is very close to the chord ...
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1 vote
1 answer
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Points on triangle median forming isosceles triangles

outline Let $\triangle ABC$ be a triangle and M the center of $\overline{AB}$. On the median $\overline{CM}$ a point D is chosen so that $\overline{CD}=\overline{AD}$ and a point E so that $\overline{...
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The incentre of triangle lies on original circle

Prove that if the tangent lines from A are drawn to the circle, the incentre of triangle ABC lies on the original circle
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3 votes
1 answer
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Prove triangle GIH is isosceles...

The geometry questions goes like this: Let A, B, C and D be four distinct points such that line segments BD and AC are of equal length. let E and F be the midpoints of line segments BC and AD ...
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Can a triangle be constructed such that

Can a triangle be constructed such that the cosines of its angles are $3/5, 5/13, 7/13$ There is no such triangle possible is the answer but how to prove it. Please help.
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min distance in a triangle

I have two questions that I have no idea where to begin. Given a triangle ABC, find a point M inside the triangle (it can be on the edges) such that AM+BM+CM in minimum. Given a triangle ABC, find a ...
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3 votes
1 answer
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Proving that $\angle MAC = 30^{\circ}$

In $\Delta ABC$, $M$ is an interior point such that, $MB=MA$, $MC=CB$, $\angle CBA = 2 \angle BAC$. Prove that $\angle MAC = 30^\circ$ I was able to solve this question using trigonometry here's my ...
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Maximum value of $ \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} $ in triangle is... [duplicate]

I was asked to prove the given inequality in a triangle Prove that $\displaystyle \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} < 2 $ As one of the side approaches to zero the remaining two sides ...
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In a triangle what is the minimum value of $ \frac{a}{c+a-b} + \frac{b}{a+b-c} + \frac{c}{b+c-a} $ [duplicate]

The given question was, In a triangle what is the minimum value of $ \displaystyle \frac{a}{c+a-b} + \frac{b}{a+b-c} + \frac{c}{b+c-a}? $ The answer for this question was found to be $3$. I tried by ...
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4 votes
1 answer
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In an equilateral triangle, infinite line segments connect a vertex to the opposite side. If the product of the lengths converges and >0, what is it?

In an equilateral triangle, line segments connect a vertex to $n$ uniformly distributed points on the opposite side, including points at the ends. Assuming $\lim_{n\to\infty}\text{(product of lengths ...
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12 votes
1 answer
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I made a pretty cool formula does it exist anywhere else?

So I’ve made a formula that takes 2 angles and a base and depending on which angle you put in first gives you one of the 2 other sides and by using certain parts of the formula and it’s inverses you ...
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5 votes
2 answers
101 views

find an equation satisfied by the three bisectors of a right triangle

let $a,b,c$ be the sides of a generic triangle and $\alpha, \beta, \gamma$ the bisectors of the respective opposite angle. It is well know that, for example, $\alpha$ divides the side $a$ in two ...
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3 votes
1 answer
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Prove that the line is parallel

Let $ABC$ be a non-equilateral triangle and $\omega$ be the inscribed circle of triangle $ABC$, which touches sides $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$, respectively. Let $G$ be the ...
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What does it mean when a question says that F and B lie on the same side as the line through A C?

I came across this question: In the triangle $ABC$, $\angle BAC = w$ and $\angle CBA = 2w$, where $2w$ is acute, and $BC = x$. Show that $AB = (3 + 4 \sin(w))x$. The point $D$ is the midpoint of $AB$...
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4 votes
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How can I use nine-point circle to solve this concyclic problem

I saw this problem on the Discord Math channel. H is the orthocenter of △ABC. D, E and F are the foot of the altitudes of △ABC passing through A, B and C respectively. Lines EF and BC intersect at R....
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How to find the angles needed to determine the bend of an arm required to make a hand touch a target?

Sorry for the confusing title, I'll do my best to describe the issue I'm having. This is for a game I'm making but I'm asking the question here because it's not a programming question - it is a math ...
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-2 votes
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How Can I Calculate Integer Points Inside Of A Triangle? [duplicate]

For example I have a triangle and I know all of the coordinates of the points and angles. How can I find integer points inside of the triangle ? Thanks in advance !
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2 answers
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Find length of edge $c$ given the lengths of $a, b$ and $r$ (the radius of circle).

I made some problem and i get stuck to solve it! The diagram for the question is: : Find the length of edge $\enspace\pmb{c}\enspace$ in terms of $\enspace\pmb{a}$,$\enspace\pmb{b},\enspace$ and $\...
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6 votes
2 answers
256 views

$\triangle ABC$ has circumcenter $O$; $BO$ and $CO$ meet $AC$ and $AB$ at $D$ and $E$. If $\angle A=\angle EDA=\angle BDE$, show they are $50^\circ$

There is surely a purely geometrical solution to this problem, but none forthcoming so far! Trigonometry confirms the result, however the quest is to solve this geometrically, almost certainly ...
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How to find Area of Triangle when you mess up?

Orginal Question like this one So I thought the base was 32 feet and the height was 16 feet but when I did 1/2 times 32 times 15 and got 265 inches squared. What have I done wrong? Problem I don't ...
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What Angle Do I Need to Know?

I am utterly confused on what I'm supposed to do cause my teacher says to use what we know about supplementary and complementary angles to find the answer but $126° + 58°$ would equal $184°$ which ...
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Angle chasing proof

Given the following diagram, how can we prove angle CDA is 2x. We are told CB = AB. How do we prove this without using alternate segment theorem? I can't seem to do it. Problem:
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-4 votes
2 answers
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In $ΔABC$, if $(a+b+c)(a−b+c)=3ac$, then what can be say about the angles of the triangle?

In $ΔABC$, if $(a+b+c)(a−b+c)=3ac$, then which of the following is $\color{green}{\text{True}}$? $\angle B=60^\circ $ $\angle B=30^\circ $ $\angle C=60^\circ $ $\angle A + \angle B=120^\circ $
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1 answer
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Finding the side lengths of a 45 degree triangle with shared hypotenuse.

Right triangle A with hypotenuse 1 and sides x and y is graphed at the origin. Right triangle B is graphed so that it shares a hypotenuse with triangle A and it's sides w and z are parallel to the ...
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Decompose a $2\times 2$ matrix to a combination of rotation matrices

The background I encounter this problem when I try to analyze the planar transformation of a 2D triangle. We ignore the translational shift in this problem. Consider a 2D triangle whose edge vectors ...
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1 vote
2 answers
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Calculate third point of Isosceles triangle given two points and angle

In this image: (Just for annotations.. The actual triangle can be pointing to any direction) I know the coordinates of "red" base points and the "blue" vertex angle $\beta$, and I ...
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3 votes
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Show $\frac{FA^4}{AB^2}+\frac{FB^4}{BC^2}+\frac{FC^4}{CA^2}\geq\frac{FA^3+FB^3+FC^3}{FA+FB+FC}$ for $F$ the Fermat-Torricelli point of $\triangle ABC$

I have been working on this journal problem for my Undergraduate Summer Research for a couple weeks now and I'm stuck. The problem is: Let $\triangle ABC$ be a triangle with its greatest angle less ...
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1 vote
1 answer
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Sum of "angles" of a 3D tetrahedron

We know that the sum of angles of a triangle equals the straight angle (180 degrees). Can we convert a 2D theorem to 3D? e. g. We can generalize the triangle to a tetrahedron, angles of the triangle ...
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2 votes
1 answer
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Finding angle in circle to produce equal areas

I have a circle that is divided into 4 quadrants with a vertical and a horizontal axis. The center of the circle (where the axes cross) is point b. The top of the vertical axis is point d. On the ...
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2 votes
0 answers
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Experimenting with code when generating the Sierpinski triangles resulted in a beautiful picture of two birds kissing each other

So I recently found out about the Sierpinski triangles and decided to code it up myself because it seemed like a fun thing to do. The original algorithm is as familiar: Take three points in a plane ...
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1 vote
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Basic Geometry, extract relation between sides and angles

I am asked to extract the geometric relation from this figure: $ABC$ is not a right triangle. angle $A=\theta_1+\theta_2$. The only thing I thought about is the cosine rule: $$BC^2=x^2+z^2-2xz\cos(\...
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Usage of similar triangles

This is typically a general question about when we could use similar triangles in real life. Googling this question let me understand that it is very possible to use them for trees, buildings heights ....
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2 votes
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What is the specific term for an Isosceles triangle where the legs are longer/shorter than the base?

I'm shocked that I couldn't find an answer to this anywhere, but I have a situation where I have to categorize isosceles triangles by whether their legs are (individually) longer or shorter than their ...
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1 vote
0 answers
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How to find the coordinates of the vertices of an equilateral triangle inscribed in a given circle?

Let $C = (a, b)$ be any given point in the plane, and let $r$ be any given positive real number. Then how to find the coordinates of the vertices of an equilateral triangle inscribed in the circle $$ \...
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2 votes
2 answers
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Understanding a 'geometrical proof' of irrationality of √2

I had been having trouble understanding a proof of the irrational nature of √2. I found this proof in the first page of the foreward to 17 theorem provers of the world where a 'geometrical proof' (is ...
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1 vote
1 answer
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Using geometry to prove that $\tan(\alpha)=\frac{\sin(\alpha)}{\cos(\alpha)},\sec(\alpha)=\frac{1}{\cos(\alpha)},...$

https://www.desmos.com/calculator/6sdbz1iahd Let $\alpha$ be the angle $XOP$. We know that: $\sin(\alpha)=PA$, and $\cos(\alpha)=PB$ $\tan(\alpha)=PC$, and $\cot(\alpha)=PD$ $\sec(\alpha)=OC$, and $\...
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0 votes
3 answers
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Solving angle relationships in triangles with limited information

What is the relationship between angles $\angle abd$ and $\angle acd$, when line $bc=\frac{1}{2}$ and line $ac=30$? b c and d are in a straight line, and point d is a right angle The answer should ...
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Angles of isosceles trapezoid given bases and height

Couldn't find this specific question so wanted to know the formulae for finding the interior angles for a isosceles trapezoid given both bases, height (and area, if needed). This website seems to be ...
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How to find an equivalent angle between $-\pi$ and $\pi$? [closed]

Let's say we have an angle such as $270$ degrees or $-5892$ degrees, or similarly in radians. How do we convert it to its equivalent value between $-\pi$ and $\pi$?
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Geometry of Right angled isosceles triangle

So I was solving problems on Pythagoras' Theorem and I started wondering about the following: Imagine we have a triangle $ABC$ right-angled at $A.$ So, we can say that $\angle B = \angle C = 45.$ Now ...
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1 answer
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I need help in determining the proper shape of a right triangle with an inscribed square

Is it possible to have a right triangle like in the image, having $BD = CD$ (or $f=e$) and $e \ne d$? Figure 1 I think that it’s only possible to have $BD=CD$ if it (Figure 1) is an isosceles right ...
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3 votes
0 answers
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Locating a point within a triangle with given conditions on distances to vertices

You're given $\triangle ABC$, with known sides, find the location of point $P$ such that $ PB = k_1 PA $ and $ PC = k_2 PA $ where $k_1 \gt 0, k_2 \gt 0$ are given constants. So for this, I ...
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4 votes
0 answers
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Geometry of a diagonal in a flag

Problem I came across the flag of Trinidad and Tobago and it got me thinking about the geometry of that diagonal. Picture below. If you look at the diagonal, you'll see it doesn't just go from one ...
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