# 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 &...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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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 $... • 2,321 3 votes 3 answers 175 views ### sin(A + B) + sin(B + C) + cos(A + C) = 3/2. Find each angle. I'm given the fact that in$\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$and ... 3 votes 1 answer 61 views ### Connecting two points inside the Koch snowflake without getting too close to the boundary Let$\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 ...
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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 ...
### 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 ...