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Questions tagged [triangle]

For questions about properties and applications of triangles.

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1answer
27 views

Calculating the annulus of a sphere with a differential change in theta

Consider the following object: I want to calculate the area of the annulus. The annulus is within the region of $$ \theta $$ and $$ \theta + d\theta $$ The answer of the area of the annulus is ...
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2answers
90 views

Geometry problem (proving a relation between sides of a triangle) [duplicate]

Let $ABC$ be a triangle with unequal sides. The medians of $ABC$, when extended, intersect its circumcircle in points $L, M, N$. If $L$ lies on the median through $A$ and $LM = LN$, prove that: $2BC^{...
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1answer
48 views

Is there any natural number triangle whose inscribed circle's radius is $1$ except length $(a,b,c)=(3,4,5)$?

Is there any natural number triangle that inscribed circle's radius is $1$ except length $(a,b,c)=(3,4,5)$? I found that there are no right triangle except $(3,4,5)$. Thm. There are only one ...
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2answers
25 views

Condition for similar triangles in complex plane

If $z_1,z_2,z_3$ and $z_1',z_2',z_3'$ are the vertices of similar triangles, then $$\begin{vmatrix}1&1&1\\z_1&z_2&z_3\\z_1'&z_2'&z_3'\end{vmatrix}=0$$ Where does this ...
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3answers
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The plane $\frac{x}{2}+\frac{y}{4}+\frac{z}{3}=1$ intersects the $x$, $y$, and $z$ axes at points $P$, $Q$, and $R$. Find the area of $\triangle PQR$.

The plane $\frac{x}{2}+ \frac{y}{4} +\frac{z}{3}=1$ intersects the $x$, $y$, and $z$ axes at points $P$, $Q$, and $R$. Find the area of $\triangle PQR$.
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What is the area of $ABCD$ parallelogram where $E$ is mid-point of BC and the area of $BEC$ is 126?

$ABCD$ is a parallelogram. Point $E$ divides $BC$ into two equal lengths. If the area of $BEF$ is 126, what is the area of $ABCD$? Source: Bangladesh Math Olympiad 2017 Junior Category. I can not ...
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Does $\triangle ABC$ exist such that $\triangle ABC \sim \triangle DEF$, with $D, E, F$ being the incentre, centroid, orthocentre of $\triangle ABC$?

Question: Does $\triangle ABC$ exist such that $\triangle ABC \sim \triangle DEF$, with $D, E, F$ being the incentre, centroid, orthocentre of $\triangle ABC$, resp.? For such a triangle to exist, ...
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Simplifying calculation of angle between planes of two triangles with a shared edge

I'm working on a problem that involves a large triangular mesh which is used to represent a scalar 2D field, and as part of this I need to find the sin of the angle between the planes of pairs of ...
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1answer
46 views

What is the area of the quadrilateral $ADEC$ in $ABC$ right triangle in the following diagram?

In the right angled triangle $ABC$, $\angle A = 90^\circ$, $AB=8$, $AC=6$, $BC = 10$. $D$ is a point on $AB$ in such way that if a perpendicular $DE$ is drawn on $BC$ from $D$ then $BE = 4$. What ...
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3answers
34 views

sides of a right-angled triangle

$$(2n + 1)^2 + (2n^2 + 2n)^2 = (2n^2 +2n +1)^2$$ It can be used to generate infinitely many sides of right-angled triangles with integer lengths by putting values of $n = 1, 2, 3, ... $ I wanted to ...
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2answers
108 views

Apex angle of a triangle as a random variable

I am not an expert in Probability Theory and I apologize if I make some mistakes in "translating" the following problem into the language of random variables. Any help also to improve the formulation ...
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1answer
42 views

Construct Triangle given bisectors and circumcircle

suppose we have three concurrent lines $g,h,k$ in the Euclidean plane which meet at a point $P\in g\cap h\cap k.$ Moreover, let $K$ be some circle with center $P$ and some radius $r>0$. I would ...
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1answer
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What is the smallest number of $45^\circ$–$60^\circ$–$75^\circ$ triangles in non-trivial substitution tiling?

Let base = $45^\circ$–$60^\circ$–$75^\circ$ triangle. Over at What is the smallest number of bases that a square can be divided into? it was determined that 23 base were needed to make a $45^\circ$–$...
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0answers
20 views

Fourier Series of Triangular waveform

Fourier Series of Triangular waveform this is the solution of Fourier series of a triangular waveform from the book Circuits and Networks: Analysis and Synthesis by Shyammohan S. Palli. In this ...
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1answer
68 views

Prove that $\frac{BC}{AH}×\frac{CA}{BH}×\frac{AB}{CH}$ = $\frac{BC}{AH}+\frac{CA}{BH}+\frac{AB}{CH}$, where $H$ is the orthocenter of $\triangle ABC$

It is an isolated problem which has a lack of context. I found that problem in a magazine which was stated such as below: In $\triangle ABC$, three altitude lines $AE$, $BF$ and $CD$ dropped from ...
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1answer
86 views

Show that no matter how $12$ points are put on a plane, there are $3$ among them forming an angle not greater than $18^o$.

Problem : Show that no matter how $12$ points are put on a plane, there are $3$ among them forming an angle not greater than $18^o$. I am not getting any ideas in solving this problem. So, there ...
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0answers
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Trigonometry right angled triangle bearing question [closed]

I don’t know how to do part b of question 16 from the picture that I have inserted, please insert a clear explanation as well. The answer on the book is 66 degrees. Thank you!
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Spherical triangles and congruence criteria

I read from different sources that the usual criteria of congruence of triangles work for spherical triangles. However it seems to me that there is a counterexample. Consider two poles on the sphere A ...
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1answer
73 views

Is the Pisot Triangle series known?

The Kepler triangle is built with powers of $\sqrt\phi$ to make a right triangle. The supergolden ratio can make a 120° triangle. It turns out that most Pisot numbers (Mathworld, Wilkipedia) 1 to 4 ($\...
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Is there something substantial behind this solution technique?

In this video, the two variable equation $(1 - \alpha - \frac{\gamma}{2})\vec A + (\frac{\alpha}{2} - \frac{\gamma}{2})\vec B + (-1 + \frac{\alpha}{2} + \gamma)\vec C = \vec 0$ is solved by ...
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0answers
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Prove that $z_1,z_2,z_3$ with equal, non-zero modulus, are vertices of an equilateral triangle if and only if $\sum_{cyc}|z_1-z_2|(z_1+z_2) = 0 $

Prove that $z_1,z_2,z_3 \in \Bbb C$ , distinct, with equal, non-zero modulus, are vertices of equilateral triangle if and only if $\sum_{cyc}|z_1-z_2|(z_1+z_2) = 0 $ I tried dividing by $z_3|z_3|\...
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2answers
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What is the value of $x+y$ if x and y are co-primes and $PR=\dfrac{x}{y}$ in the diagram?

$ST$ is the perpendicular bisector of $PR$ and $SP$ is the angle bisector of $\angle QPR$. If $QS=9cm$ and $SR=7cm$ then $PR=\dfrac{x}{y}$ where x, y are co-primes. $x+y$=? Source: Bangladesh ...
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3answers
60 views

Point $X$ is on the circumference of the circle $PQR$ and $PY$ is a perpendicular on $XR$. What is the value of $QX + XR$?

$\triangle PQR$ is an isosceles triangle where $PQ = PR$. Point $X$ is on the cicumcircle of $\triangle PQR$ such that it is in the opposite region of $P$ with respect to $QR$. $PY$ $\perp$ $XR$ and $...
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2answers
48 views

Find the value of angle using elementary geometry rules

In $\triangle ABC$ with base $AC$, $\angle C$ = $46^\circ$ and $AC$ is extended to point $D$. $E$ is a point on $AB$ and $DE$ is joined. Given that $AB=AD=DE$. Find $\angle ABC$.
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Bijection between spherical and planar triangle surfaces

I subdivide a unit sphere, centered at origin, onto 20 spherical triangles. For the sake of argument let's take one such triangle $Ts$, in $\mathbb{R}^3$, that has vertices $Normalize(-1,0,g), ...
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3answers
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Find the $\angle ACB$ of $\triangle ABC$.

If $PC=2BP$, $\angle ABC= 45^\circ$, and $\angle APC=60^\circ$, find $\angle ACB$. All solutions are acceptable but please try solving using reflection of point $C$ through the line segment $AP$. I ...
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2answers
87 views

Find the length of a triangle's side

I have the following triangle and I'm supposed to find the value of x. First thought that came to mind is to use the the following tangent equation $$\tan(y)=\frac{x}{27}$$ and $$\tan(19+y) = \frac{...
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2answers
50 views

What is the area of $\triangle ABC$ where $\triangle ADC$ is cyclic, point $P$ is on the circumference and $AD = AP$?

$\triangle ABC$ is a right angled triangle. The perpendicular drawn form $A$ on $BC$ intersects $BC$ at point $D$. A point $P$ is chosen on the circle drawn through the vertices of $\triangle ADC$ ...
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1answer
33 views

Altitude of a Triangle: Showing indeed perpendicular

$\textbf{Question:}$ Why can we drop a perpendicular $h$ between the two points along the base of the triangle below where $\alpha, \beta\leq 90^\circ$? The reason I'm wondering this is because often ...
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1answer
31 views

Prove That Area of Isoceles Triangle in an Ellipse is Maximum When Vertex On The Major Axis Lies On The Line Of Symmetry of the Triangle?

This is one of 101 classes questions whose solutions can be easily found on google, but most of the solutions assume without giving any proper line of reasoning that to maximize area (unique)vertex on ...
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1answer
76 views

A triangle has sides $x+15$, $2x+15$, and $4x+15$. Find the range of possible values of $x$.

In a triangle with legs $x+15$ and $2x+15$ and longest side $4x+15$, the side lengths are given below in terms of a real-valued variable $x$. Find the range of all possible values of $x$, writing your ...
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3answers
49 views

If a line through the centroid G of triangle ABC meets AB in M and AC in N then prove that AN.MB +AM.NC = AM.AN both in magnitude and sign. [closed]

If a line through the centroid $G$ of $\triangle ABC$ meets $AB$ in $M$ and $AC$ in $N$ then prove that $$AN.MB +AM.NC=AM.AN$$ both in magnitude and sign.
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What is the value of the length $BD$ in a cyclic $\triangle BCD$?

So, here is my question: In $\triangle BCD$, $BC$ = $5$ and $G$ & $H$ are two points on $CD$ such that $CG$ = 1, $GH$ = 3 and $HD$ = 2. $BG$ and $BH$ intersect the circumcircle of $\triangle ...
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1answer
77 views

EGMO Problem 3.20 (BAMO 2013/3)

The problem statement follows: Let $H$ be the orthocenter of an acute angle triangle $ABC$. Consider the circumcenters of triangles $ABH$, $ BCH$, and $CAH$. Prove that they are the vertices of a ...
3
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1answer
45 views

Equality in triangle obtuse

$m_1, m_2, m_3 $ are sides-lengths of a triangle such that $m_1\sqrt{m_1}+m_2\sqrt{m_2}=m_3\sqrt{m_3}$. Prove that this triangle is an obtuse-angled triangle. I don't have idea make run this ...
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1answer
394 views

Patterns in inequalities of triangle involving angles.

I was reading this page and wondered as why, inequalities for $\cos A$ (with argument $A$) become the same inequality for $\sin\frac{A}{2}$ (with argument $\frac{A}{2}$), similarly for $\tan$ and $\...
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1answer
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Calculating length of triangle sides in trapezium

My younger brother has this mathematical problem to solve, and he came to me for help. At first I thought I could solve it by simply applying the Pythagorean theorem, but there seems to be more to it. ...
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3answers
55 views

Find the value of $CE$. [closed]

$ABCD$ is a square of side 1, the value of $FE$ is 1 and the points $A$, $C$, $E$ are collinear, as well as $B$, $F$, $E$. The question is to find the value of $CE$. My teacher gave me this challenge ...
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2answers
36 views

A height in isosceles triangle

isosceles triangle with height and distances $$M \in AB$$ $$MM_1 \bot AC, M_1 \in AC$$ $$MM_2 \bot BC, M_2 \in BC$$ $$AH \bot BC, H \in BC$$ I have to show that: $$MM_1 + MM_2 = AH.$$ I've ...
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4answers
69 views

What is the value of $m+n$ if $\frac{m}{n}$ is the radius of the smallest of the three circles?

Circles of radii 5, 5, 8 and $\frac{m}{n} $(the smallest circle) are mutually externally tangent to all circles, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Source: ...
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1answer
31 views

Given a strip of 7 small triangles each with an area of 1, what is the area of the created trapezoid below the strip?

All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the 7 smallest triangles has area 1, and $\triangle ABC$ has area 40. What is the area of ...
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1answer
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In $\triangle ABC$, $D$ is an exterior point such that $AC = CD$ and $CE$ is parallel to $AF$. Find the area of $ABDF$.

In $\triangle ABC$, $CB$ is extended upto $D$ so that $AC$ = $CD$. An angle $\angle DCE$ is drawn at point $C$ so that is equal to $\angle CAB$ and $AB$ meets $CE$ at $I$.$E$ is such an external point ...
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1answer
85 views

Solving for the value of $ \angle CEB$ - $\frac{1}{4}$ $\angle CBA$ where $E$ is an exterior point of $\triangle ABC$

From point $A$ of $\triangle ABC$, a line $AD$ parallel to $CB$ is drawn so that $AD=AB$. From point $B$, a line parallel to $AC$ is drawn so that it can satisfy the term of $BE=BC$. Point $D$ and $E$ ...
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2answers
87 views

Maths exam question on consecutive Pythagoras triplets

Our 13 year old daughter brought home this maths question that she was asked in a "maths challenge" last week: Q. The values of the adjacent and opposite sides in a right angle triangle of lengths 3, ...
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0answers
24 views

Finding the whole triangle information by one point.

I wonder if there's any way to blend two (or more) RGB colors in a reversible way? I mean, imagine we have an RGB pixel (R: 55, G:35, B:255), and we need to extract the two other RGB pixels that ...
4
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3answers
74 views

In a right angled $\triangle ABC$, $DE$ and $DF$ are perpendicular to $AB$ and $BC$ respectively. What is the probability of $DE\cdot DF>3$?

In a right angled $\triangle ABC$, $\angle B = 90^\circ$, $\angle C = 15^\circ$ and $|AC| = 7.\;$ Let a point $D$ be taken on $AC$ (according to the estimated diagram which is drawn below) and then ...
3
votes
2answers
72 views

Computing sum of lengths of legs of a right triangle

Consider a right triangle $\triangle ABC$, where the right angle is $\angle A\hat CB$, as in the picture below. Let $\alpha=\angle A\hat BC$. Problem: To determine the sum of the lengths of the legs ...
1
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1answer
54 views

Finding the length of $HD$ in a cyclic $\triangle BCD$.

Source: Regional Math Olympiad of BD Let $\triangle BCD$ be a right angled triangle where $\angle BCD$ = 90$^\circ$. $FG$ is perpendicular to $BD$ from any point $F$ on $CD$. Line $BF$ intersects ...
3
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2answers
91 views

Perimeter of orthic triangle

If $DEF$ is the orthic triangle of $\triangle ABC$, then prove that $$\frac{\text{Perimeter of }\triangle DEF}{\text{Perimeter of }\triangle ABC} = \frac{r}{R} $$ where $r$ and $R$ are the inradius ...
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1answer
50 views

A simple euclidean geometry problem of angles of a triangle using linear equations [duplicate]

I am having problems solving this problem: Using only basic geometry is easy to go here: And propose 4 equations: $$ x+y+70=180$$ $$x+w+40=180 $$ $$u+y+50=180 $$ $$u+w+20=180 $$ And it doesn't make ...