Questions tagged [triangles]

For questions about properties and applications of triangles.

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sufficient condition for a triangle to exist (and fibonacci numbers)

Let $n\in\mathbb{N}\setminus\{1,2\}.$ Let $a_1,a_2,\ldots,a_n$ be $n$ (not necessarily distinct) real numbers each in the interval $(1,F_n),$ where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Show ...
idk's user avatar
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what is the trilinear coordinates of the intersection between these 2 circles

(I really searched on the internet, I found NOTHING) In a triangle, we can draw the 9-point Euler circle, and the 3 excircles of that triangle. there always is an intersection point between the Euler ...
Pierre Carlier's user avatar
1 vote
0 answers
49 views

Does this shape have a name? (A 'spherical circular triangle' ???)

This shape is formed by 3 'small' circles on the surface of a sphere, each touching the other 2. On a plane, the shape is called a 'circular triangle' (refer to Wikipedia). In this particular example ...
Parsley's user avatar
  • 11
0 votes
1 answer
40 views

Problem in a triangle

$ABC$ is an isosceles triangle so $AB=BC$, $D$ is the midpoint of $BC$ so $BD=DC$ and $BED$ and $DFC$ are isosceles triangles such that $BE=ED=DF=FC$. We know that $BED$ and $DFC$ are isosceles ...
Birgitt's user avatar
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1 vote
2 answers
133 views

How special is $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$?

The well-known identity for complex points forming an equilateral triangle reads $$z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$$ I have a doubt concerning the uniqueness of this identity Is $...
rgvalenciaalbornoz's user avatar
0 votes
1 answer
194 views

Prove that the distance $d$ from point $A$ to plane $(DNC)$ verifies the relation $d< \frac{AB+3AD}{6\sqrt{2}}$

the question In the triangle $ABC$ we consider $(AM$ the bisector of the angle $\angle A$ so that $MB=3MC, M\in (BC)$ and $N\in (AB)$ so that $BN=2NA$. On the plane of the triangle $ABC$, the ...
IONELA BUCIU's user avatar
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1 vote
1 answer
60 views

Area of a triangle in Lockhart's Lament

In the essay Lockhart's Lament (page 4), the author describes a proof for the standard area of triangle $(bh)/2$ by enclosing the triangle in a rectangle and chopping the rectangle into two (perhaps ...
MathArt's user avatar
  • 113
-4 votes
0 answers
23 views

Hypotenus of a scalene triangle [closed]

How do I find the length of the line AC in the triangle ABC, using Microsoft Excel, if angle ABC is 90, BCA is 50, BAC is 40, line BC is 7 cm, and length of AB is unknown?
Vyas's user avatar
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0 votes
1 answer
20 views

How to construct $\Delta ABC$, given the angle at $C$ , $\gamma$ the median drawn fom $C$, $t_c$, and the angle at the point $B$, $\beta$? [closed]

I've gotten stuck again, can't seem to figure out what I'm supposed to construct first. $\Delta ABC$ with $\gamma$, $\beta$ and $t_c$ marked. Next to it text reads: $\gamma$ = $\angle ACB$, $\beta$ = $...
WhyWho's user avatar
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0 votes
1 answer
70 views

How can you construct a right triangle, ABC with a right angle in C, if you are given the hypotenuse, c, and the altitude of the point C? [closed]

How can you construct a right triangle, $\triangle ABC$ with a right angle in $C$, if you are given the hypotenuse, $c$, and the altitude of the point $C$? I know it's very basic but I just can't seem ...
WhyWho's user avatar
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-1 votes
0 answers
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Precalculus-level math help: is this solvable? [closed]

$\triangle{A_0B_0C_0}$ has $\angle A_0 = 40^{\circ}$, $\angle B_0 = 60^{\circ}$, and angle $\angle C_0 = 80^{\circ}$. It is rotated about vertex $A_0$ clockwise by some angle $\alpha$ to get $\...
Mintylemon66's user avatar
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51 views

How to solve the limit of the following trigonometric functions?

How can i calculate limits of $\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{x \to + \infty } \cos \left[ {\left( {a + i\varepsilon } \right)x} \right] = a$, $\mathop {\lim }\...
Tom's user avatar
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5 votes
1 answer
86 views

Can a triangle up to isometry shatter seven points?

From Wikipedia: The class [of sets] $C$ shatters the set $A$ if for each subset $a$ of $A$, there is some element $c \in C$ such that $a = c\cap A$. In other words, $C$ shatters $A$ if for every ...
RavenclawPrefect's user avatar
2 votes
1 answer
92 views

Prove that the triangles $VAC, EAV$ are similar if and only if $\angle EVO=30°$.

The question The regular quadrilateral pyramid $VABCD$ has the vertex $V$. Let $M$ be the middle of the edge $AD$ and $E$ be the point of intersection of the lines $AC$ and $BM$. Prove that the ...
IONELA BUCIU's user avatar
  • 1,261
-2 votes
0 answers
72 views

Circles and Angles

$H$ is the orthocenter of acute $\triangle ABC$ and the extensions of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$ intersect the circumcircle of $\triangle ABC$ at $A'$, $B'$ and $C'.$ We ...
Goku's user avatar
  • 15
2 votes
2 answers
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A tournament is acyclic if and only if it has no triangles

A tournament is a directed graph where between any two distinct vertices there is either the edge (u,v) or the edge (v,u) (one of them only). I have not come across a proper explanation on why the ...
Yavuz Bozkurt's user avatar
0 votes
2 answers
68 views

Creating a relevant right triangle to evaluate $\sec\left(\arctan\frac{4}{3}\right)$

I am trying to solve the following: $$\sec\left(\arctan\left(\frac{4}{3}\right)\right)$$ The problem tells me to use a relevant right triangle, but I am curious as to if I need to create a right ...
MSM's user avatar
  • 25
-1 votes
1 answer
109 views

Prove of inradius of a right angle triangle. R²= R1²+ R2²

A right-angle triangle ABC is present whose right angle is BAC. One perpendicular from point A is taken on BC which is AD. Then in triangle ABC, triangle ABD, and triangle ADC, three inscribed circles ...
Samagata Banerjee's user avatar
0 votes
1 answer
37 views

Explanation to how the length between 2 centers is adjusted as angle changes [closed]

I'm working on a project, that takes 2 similar elements and allows you to adjust the angle of the center-line by grabbing the right side and moving it up or down. In this scenario, lets use the key: ...
knocked loose's user avatar
1 vote
2 answers
117 views

What is the quantitative relationship between $∠BAF$ and $∠CHG$ and the line segment that is equal to $AF$?

Question: As shown in the diagram, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension ...
rubycon's user avatar
  • 13
3 votes
0 answers
164 views

A triangle is cut into several triangles, one isosceles (not equilateral) and the rest equilateral. Determine the angles of the original triangle.

This question has been taken from III GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN: A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all ...
curious's user avatar
  • 31
1 vote
0 answers
38 views

Is the tetrahedron necessarily mirror-asymmetric?

I was shocked to watch Anton Petrov's latest video, "Wow, Incredible Evidence That Universe Is Not Symmetric After All", where he says that the Tetrahedron is the simplest object that is not ...
Miss_Understands's user avatar
2 votes
3 answers
114 views

Problem with triangle in a circle

We are given that $ABC$ is equilateral so $AB=BC=CA$, and that the length of the circle is $24\pi$. What is the area of the triangle? Since $24\pi=2\pi r$ we get that $r=12$ and $OB=OC=OA=r$; $OAB$, $...
Birgitt's user avatar
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0 answers
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$ABC$ is a triangle with $AB = \sqrt{3}/2$, BC =1, and $B$ is a right angle. $PQR$ is an equilateral triangle with sides $PQ, QR$, and $RP$ passing [closed]

$ABC$ is a triangle with $AB = \sqrt{3}/2$, BC =1, and $B$ is a right angle. $PQR$ is an equilateral triangle with sides $PQ, QR$, and $RP$ passing through the points $A, B$, and $C$ respectively and ...
IMRUL KAYES's user avatar
4 votes
1 answer
39 views

Discern how many triangles are possible given two sides and a non-included angle

For the longest time, I always thought you could only determine a triangle, having only two sides, if you also had the included angle. Recently I solved a trigonometry problem where I was given $2$ ...
zlaaemi's user avatar
  • 1,009
1 vote
1 answer
61 views

Problem to prove a point lies on a circle. [closed]

In $\triangle ABC$ it is given that $\angle ABC=60$. Let the perpendicular bisector of $AC$ meet angle bisector of $\angle B$ at $X$. Prove that $X$ lies on the circumcircle of $\triangle ABC$. I ...
Adhvik's user avatar
  • 61
0 votes
2 answers
62 views

Geometric question related area of sectors in circles

The question is as such: Triangle ABC is a equilateral triangle of side length 8 cm. Each arc shown in the diagram is an arc of a circle with the opposite vertex of the triangle as its center. The ...
koiboi's user avatar
  • 157
-1 votes
1 answer
78 views

40-50-90 Triangle with Cosine Equation

Triangle DEF contains right angle E. If angle D measures 40° and its adjacent side measures 7.6 units, what is the measure of side EF? Round your answer to the nearest hundredth. So far, I have angle ...
Lizzy's user avatar
  • 3
0 votes
1 answer
33 views

Volume of an n-dimensional isosceles pyramid

I was looking at the following question (link). To summarise, the aim is to find, for standard uniform distributions $U_i$, $$P(U_1+\ldots+U_n \leq 1)$$ for $0\leq t\leq1$. A nice proof by induction ...
user19904's user avatar
0 votes
1 answer
47 views

Finding out the coordinates of the right angle in a right triangle, knowing only the other two coordinates and angle relative to the plane [closed]

I'm faced with the situation as presented on this picture. I know the coordinates of $A$ and $B$. And I know that the vectors $AC$ and $CB$ lay in a $45^{\circ}$ angle relative to the plane (as shown ...
AnB's user avatar
  • 3
6 votes
1 answer
88 views

Determining the circumdiameter of a triangle, given the distance from a vertex to the orthocenter and the length of the opposite side

Problem I'm trying to solve: Given triangle $ABC$ inscribed in a circle a shown. The altitudes of the triangle meet at point $P$. $AP=21\text{cm}$ and $BC=20\text{cm}$. What is the diameter of the ...
d0uble_a_b4ttery's user avatar
1 vote
3 answers
71 views

In a circle of radius 5 , AB and AC are 2 chords such that AB = AC = 6cm. Find BC

In this question I can easily calculate the perpendicular distances of AB and AC from the centre O by the perpendicular bisector property. However , after this I am left with a quadrilateral BOFC (F ...
Krishang Rana's user avatar
0 votes
1 answer
47 views

On Ratios in Isosceles Triangles

It is known that the formula for the perimeter $P$ of an isosceles triangle on a plane is $$P=2L+B$$, where $L$ is the length of the leg and $B$ is the length of the base. Now, let us study some ...
Michael Ejercito's user avatar
7 votes
3 answers
408 views

Ratios of lengths in a triangle

Problem I'm trying to solve: My Attempt: To be honest, I don't have any clue on how to solve it and the method I'm about to give is severely overcomplicated. We first isolate triangle $\triangle BEG$ ...
d0uble_a_b4ttery's user avatar
4 votes
1 answer
181 views

Angles created by triangle median [closed]

Given that $AD$ bisects the line $BC$ into equals parts $BD$ and $DC$, does that tell us anything about the angles $x$ and $y$? We also know $\angle B$ and $\angle C$. Is all this enough info to find ...
user1078's user avatar
  • 343
13 votes
2 answers
271 views

Conjecture: Expected total area of a certain set of random triangles in a unit disk is $1/\pi$.

Choose $3n$ independent uniformly random points in a disk with perimeter $x^2+y^2=1$. Label the points $P_1,P_2,P_3,\dots,P_{3n}$ in order of increasing $x$-coordinates. Form triangles $\triangle ...
Dan's user avatar
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17 votes
3 answers
865 views

Three random points on $x^2+y^2=1$ are the vertices of a triangle. Is the probability that $(0,0)$ is inside the triangle's incircle exactly $0.13$?

Three uniformly random points on the circle $x^2+y^2=1$ are the vertices of a triangle. What is the probability that $(0,0)$ is inside the triangle's incircle? (This a variation of the question &...
Dan's user avatar
  • 21.2k
3 votes
1 answer
94 views

Can the internal angles of a triangle ever add up to 360 degrees, or more? [closed]

On flat 2d paper, a triangle's internal angles add up to 180 degrees. Drawn on the side of a 3d soccer ball, they add up to 270 degrees. Can they ever add up to 360 degrees (or even more) as the ...
Paul Uszak's user avatar
1 vote
1 answer
94 views

Angular Position of the Man

A man is standing at a distance of 60m from one tree and 54m from another tree on a playground. If the area of the triangle formed by the man and two trees is 810m², at what angular position is the ...
Techno Highway's user avatar
24 votes
3 answers
762 views

What's the area of the triangle in this geometry problem? I think I can solve it, but it's way too convoluted...

I am trying to solve this geometry problem from an exam. The exam is supposed to be 3 hours long and this is supposed to be 1 out of 10 problems. So given that, the solution should be something quick, ...
zlaaemi's user avatar
  • 1,009
1 vote
1 answer
95 views

How to find the ratio of sides of 45-45-90 triangle independently of Pythagoras theorem?

I've managed to find out the ratio of sides of $30$-$60$-$90$ triangle through the length of angle bisector, which does not depend on Pythagorean theorem at all, looking at its proof. I did it in the ...
Rusurano's user avatar
  • 686
1 vote
0 answers
47 views

Can a triangle $ABC$ be translated onto another triangle $PQR$ in multiple ways?

For example, if $ABC = (0,1), (2, 3), (4, 7)$ and $PQR = (-1, -1), (1, 1), (3, 5)$, then the only way $ABC$ can be superimposed onto $PQR$ is by a translation $1$ unit left and $2$ units down. This ...
John's user avatar
  • 1,930
2 votes
3 answers
211 views

What is the length of AR in the given triangle ABC?

I need your help with a question on triangles. It is from the entrance exam of an institute in India for students in 10th grade. In the given triangle ABC, if DP, MQ AR and ES are perpendiculars to ...
ROHIT's user avatar
  • 23
2 votes
0 answers
62 views

triangle inscribed in a triangle: Minimum length of two sides [closed]

Have been reading Shwarz's proof of the minimal permiter of a triangle inscribed in another triangle as outlined in chapter VII of Courant and Robbins' "What is Mathematics" and they assume (...
David's user avatar
  • 31
0 votes
1 answer
59 views

A rectangle's diagonal is divided into thirds by heights from opposite points... [closed]

I need help with the following triangle similarity problem: A rectangle's diagonal is divided into thirds by heights from two opposite points of the rectangle. If the length of one of the sides of the ...
Kerim's user avatar
  • 13
0 votes
1 answer
51 views

Prove that $E,F$ and $I$ are Collinear points.

Let $\Delta ABC$ be a arbitrary triangle. And $E,F$ and $I$ points such that : $$\vec{AE}=\dfrac{2}{3}\vec{AB}~,~\vec{BI}=-\dfrac{1}{3}\vec{BC}~,~ \vec{AF}=\dfrac{1}{3}\vec{AC}$$ Question Prove that $...
Ellen Ellen's user avatar
  • 2,321
3 votes
3 answers
175 views

sin(A + B) + sin(B + C) + cos(A + C) = 3/2. Find each angle.

I'm given the fact that in $\triangle ABC$, $\sin(A + B) + \sin(B + C) + \cos(A + C) = \frac{3}{2}$. I'm asked to get the angles $A, B, C$. So far what I've done is I've substituted $A+B = 180-C$ and ...
StrugglinStudent's user avatar
3 votes
1 answer
61 views

Connecting two points inside the Koch snowflake without getting too close to the boundary

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. We say $\Omega$ is a uniform domain with constant $c \geq 1$ if for any $x,y \in \Omega$ there is a rectifiable curve $\gamma : [0, l_\gamma] \to ...
Tobi's user avatar
  • 78
3 votes
2 answers
59 views

Given point in square and its distance to two vertices, how many distinct integer values are possible for the area?

Let $ABCD$ be a square. Suppose $P$ be a point strictly inside the square such that $AP = 5$ and $BP = 13$. How many distinct integer values are possible for the area of $ABCD$? The possible choices ...
MathNewb's user avatar
0 votes
1 answer
38 views

How to prove that in any triangle $ABC$, there exists a point $D$ on the longest side of the triangle, $BC$, such that $AD \perp BC$?

How to prove that in any triangle $ABC$, there exists a point $D$ on the longest side of the triangle, $\overline{BC}$, such that $\overline{AD} \perp \overline{BC}$, using only geometry. I tried ...
Mohammad muazzam ali's user avatar

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