Questions tagged [triangles]

For questions about properties and applications of triangles.

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Find geometric relation from a figure

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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If $a+b=(a\times \arctan(\hat A) + b\times \arctan (\hat B))\tan\frac{\hat{C}}{2}$, then $\triangle ABC$ is isosceles

ABC is a triangle with $A,B,C$ as the angles and $a,b,c$ as the sides opposite to the respective angles. Prove that if: $$a+b=\left(a\times \arctan(\hat A) + b\times \arctan (\hat B)\right)\tan\frac{\...
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Notable elements of any triangle ABC [closed]

Here's a link for a display of notable elements of any triangle ABC with Steiner Deltoid, Mandart Inellipse, morley's triangles, Apollonius circle, cubics, etc... https://www.desmos.com/calculator/...
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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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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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2 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 $ac=30$? Now, I know this is solvable because if we extend line $ba$ to infinity, and then ...
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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 $\pi$?

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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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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2 votes
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Locating a point within a triangle with given conditions on distances to vertices

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 translated the ...
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1 answer
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If $I$ is the incenter of triangle $ABC$ whose inradius is $r$ then how to find the inradius of triangle $BIC$ [closed]

Actually I have to find the relation between the inradii of triangles $ABC, AIB, BIC, CIA $
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How to get radii of incircles of triangles AIB,BIC and CIA where I is the incentre of triangle ABC

How to get radii of incircles of triangles AIB,BIC and CIA
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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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Is there an alternative way to completely rewrite Heron's Formula in terms of $p$ (or $p$ and the sides)?

So I was playing around with Heron's Formula and I noticed that: $\sqrt{s(s - a)(s - b)(s - c)} \equiv \frac{1}{4} \sqrt{p(p - 2a)(p - 2b)(p - 2c)}$ where $s$ is the semiperimeter of a triangle ($\...
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2 answers
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A triangle with side lengths x, x^2 and x^3

If I have a triangle with side lengths x, x^2, and x^3, what are the possible values of x expressed in interval notation?
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5 votes
2 answers
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APMO 2020 Geometry Problem | Proving lines to be concurrent

PROBLEM Let $\Gamma$ be the circumcircle of $∆ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ ...
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On a coordinate plane you are given three vertices and trying to find the area.

If the triangle given has the slope of 0 between AB then how would you solve the problem. We are trying to find the area. ex.) A=(0,0) B=(12,0) c=(6,8)
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3 votes
1 answer
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Volume of a triangular prism with 2 different bases

How do I arrive at a formula to calculate the volume of the following 3D shape? Does this shape have a proper name? It kind of looks like an irregular triangular prism with 2 similar triangles as ...
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Finding area of triangle OAC [closed]

Given that AB // OD, OA = OD = 5 cm, AH = 3 cm. OB = OC = 9 cm , angle AOD = angle BOC . Find the area of triangle OAC. Does this question solve without using similarity? How can I solve it? I am ...
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Dodecagon inscribed in a circle

A regular dodecagon ABCDEFGHIJKL is drawn in a circle radius 2 and centre O. Let P be the point where AD cuts the line OB. (a) Show that triangle ∆APB is similar to ∆OAB. I have researched online but ...
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7 votes
2 answers
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Generating Pythagorean triples where the legs are Hypotenuses of other Pythagorean triples

I know how to generate regular Pythagorean Triples given two positive integers P and Q such that $$a=2*p*q$$ $$b=p^2-q^2$$ $$c=p^2+q^2$$ where $p>q$, but I want to find scenarios where $a$ and $b$ ...
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-1 votes
1 answer
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Finding maximum area of triangle

Let $A=(3,1), \space B=(4,3), \space C=(4,4)$ be three point in coordinate plane with origin $O$. Let $D$ be arbitrary point on line segment $OC$. what is largest possible area of $\triangle ABD$. I ...
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2 votes
0 answers
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Intersection of inscribed equilateral triangles in a circle [closed]

I'm trying to find a proof for this proposition, but I can't find an initial hint. Note: This proposition comes from numerical evidence (Geogebra). My first try was to use formulas from integral ...
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Circumcenter of acute-$\Delta ABC$ is $O$. Line $AC$ intersects circumcircle of $\Delta AOB$ at point $X$, in addition to vertex $A$. Prove $XO⟂BC$.

PROBLEM The circumcenter of an acute-$\mathtt{\Delta ABC}$ is $\mathtt{O}$. Line $\mathtt{AC}$ intersects the circumcircle of the $\mathtt{\Delta AOB}$ at a point $\mathtt{X}$, in addition to the ...
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What would be the Mean shortest distance from random points in the right angled triangle to the Hypotenuse.

The problem is to find the average shortest distance of uniformly random points from the hypotenuse in a right angled rectangle. The distance d shows the shortest distance to the hypotenuse from a ...
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2 answers
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How to prove that angle bisector of right angle triangle ABC right angled at B is perpendicular bisector of third side AC. [closed]

I have tried using sine theoram, angle bisector theoram, congruency of type RHS,AA,ASA but haven't been able to do this.
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2 answers
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Why doesn't this approach to deriving the hyperbolic functions work?

This question may be the closest to what I'm looking for, but the link in the answer uses a CAS to complete the proof. I was thinking I could derive the formulas for $\sinh{x}$ and $\cosh{x}$ using a ...
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0 votes
1 answer
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Constructing equilateral triangles on the sides of a right angled triangle

Let $ABC$ be a right triangle right angled at $C$. Three equilateral triangles $BCP,ACQ,ABR$ such that $P-A,Q-B,R-C$ are points on the opposite sides of $BC,AC,AB$. Prove that $AP=BQ=CR$ I proved ...
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Right Angles in a Triangle

Let $ABC$ be a triangle and $D$ a point on its interior so that $DBC = 60°$ and $DCB = DAB = 30°$. If $M$ and $N$ are the middlepoint of $AC$ and $BC$, respectively, show that $DMN = 90°$ I know that $...
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-1 votes
3 answers
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Finding a non-equilateral triangle whose sides form a geometric sequence and whose angles form an arithmetic sequence

This question was on a math competition. Is there a triangle, which is not equilateral, whose sides form a geometric sequence and whose angles form an arithmetic sequence? If such a triangle exists, ...
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Calculating unknowns of nested triangles with common sides

I have 3 triangles joined together with common legs between them. For triangle ABC (the blue one in the diagram below) we know all its angles and the length of all it's sides. For the other two ...
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Finding a bearing angle from a triangle.

I am a computer science students and I have encountered a task in my learning resource which requires some level of math I couldn't understand. So I have a triangle and all of the known variables are: ...
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How to solve $\beta = \alpha - \arcsin(k \cdot \sin(\alpha))$ for $\alpha$?

I'm looking for an analytical expression in the form of $\alpha = f (\beta)$, so solving the equation below for $\alpha$. Is this possible? $$\beta = \alpha - \arcsin(k \cdot \sin(\alpha))$$
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3 votes
3 answers
203 views

Nearest point of the vertices of a triangle

I would like to know if the point closest to the three vertices of an equilateral triangle is the centre of its circumcircle, and if so, how to prove it. By closest point, I mean that any other point ...
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1 vote
2 answers
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Inscribe 3 equal circles in a Reuleaux/curved triangle without overlap [closed]

How exactly can I inscribe 3 equal circles into an equilateral Reuleaux triangle like the attached image without any overlap? I have the construction of the Reuleaux triangle down, drawn the altitudes,...
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1 vote
1 answer
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Why is the distance from orthocenter to vertex twice the distance from circumcenter to opposite side?

In the diagram above, $$2SP=AO$$ in description : line from orthocenter is 2 times of line from circumcenter. But I remember, someone in MSE said It's Euler line (I have read Wikipedia article but ...
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Find a point on triangle and interpolated triangle normal which points to specific point in 3D

Suppose i have triangle in 3d with vertices $v1, v2, v3 \in R^3$. Each triangle vertex has associated normal vector $ n1, n2, n3 \in R^3, ||n_i|| = 1$. In computer graphics such vectors sometimes ...
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Find angle QTS.

Given $PQ=PR, QT = 2SU$, $PS \parallel QR$ and $SU \perp QR$. $QT$ is angle bisector of $\angle PQR$. What is the size of $\angle QTS$? What I have so far: $\angle PQT=\angle RQT=\alpha$, $\angle QRP =...
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Possible triangle center associated with Apollonius circle of excircles

When I was playing around with Geogebra, I personally found a possible triangle center, but I'm not 100 % sure if my personal conjecture is true. Consider the following configuration: Let $E_A$, $E_B$...
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-1 votes
1 answer
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Find a point X inside triangle ABC such that (area AXB): (area BXC): (area CXA) = k : l : m where k, l, m are given constants. [closed]

I feel like areas are related to AX, BX and CX. But these are just my assumptions. I don't know how to solve this question but I stared at the rough figure of this question for a long time. I draw ...
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3 votes
1 answer
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169 points are chosen at random inside an equilateral triangle of perimeter 300.

I have this excercise: 169 points are chosen at random inside an equilateral triangle of perimeter 300. Prove that there are 3 of these points that determine a triangle of area at most 68. I think ...
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0 votes
1 answer
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In the figure angle AB is inscribed in thesemicircle. What is the value of angleD C A (C is the centre of the circle)? [closed]

In the figure angle, AB is inscribed in the semicircle. What is the value of angleD C A (C is the centre of the circle)? (A)61o (B)129o (C)58o (D)122o (E)121o
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1 vote
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Is it true that the sum of squares of perpendicular medians in a triangle is equal to the square of the third median?

I am aware of the fact that two medians are perpendicular in a triangle if and only if the Sum of the squares of the lengths of the sides to which these medians are drawn is $5$ times greater than ...
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Prove that $AP \cdot AQ+CP \cdot CQ=BP\cdot BQ$

Let $G$ be the centroid of $\triangle ABC$. A line $M$ through $G$ intersects the circumcircle of $\triangle ABC$ at $P$ and $Q$, where $A$ and $C$ lie on same side of $M$. Prove that $AP \cdot AQ +CP ...
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0 votes
1 answer
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Find the distance between two towers/building

Mr Gray stood on top of the CN tower and spotted a Casa lama, his angle of depression is 6 degrees. He then turned around 110 degree and found CNE. His angle of depression to CNE is about 9 degree. ...
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1 vote
2 answers
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In $\triangle ABC$, $D$ is on $AC$ such that $AB=CD$, $E$ and $F$ are midpoints of $AD$ and $BC$, and $BA$ intersects $EF$ at $M$. Prove $AM=AE$.

A triangle $\triangle ABC$ has a point $D$ on $AC$ such that $AB=CD$, and $E$ and $F$ are midpoints of $AD$ and $BC$, respectively. $BA$ intersects $EF$ at $M$. Prove that $AM=AE$. I know the ...
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