Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [triangles]

For questions about properties and applications of triangles.

A triangle is a polygon with three sides. It is an important geometric figure, because any polygon can be subdivided into triangles.

Triangles can be classified by the number of sides they have that have equal length

  • All three sides of an equilateral triangle have equal length.
  • An isosceles triangle has at least two sides of equal length.
  • A scalene triangle is a triangle that is not isosceles, that is, it has no sides with equal length.

A triangle may also be classified by describing its angles. A triangle is said to be a right triangle if it contains a right angle, and obtuse triangle if it contains an obtuse angle, or an acute triangle if all three of its angles are acute.

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Why do the interior angles of two parallel lines (made by the transversal) add up to 180 degrees?

Does anyone have some simple proofs here? I was looking for some proofs for corresponding angles are equal, but in the one i found they use this theorem that states that the interior angles of two parallel lines (made by the transversal) add up to…
excellence
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$\Delta ABC$ is right angled triangle. $AP$ and $AQ$ meet $BC$ and $BC$ produced in $P$ and $Q$ and are equally inclined to $AB$.

$\Delta ABC$ is right angled triangle at $A$. $AP$ and $AQ$ meet $BC$ and $BC$ produced respectively in $P$ and $Q$ and are equally inclined to $AB$ ($\angle BAP=\angle BAQ$). Show that $\frac{BP}{BQ}=\frac{CP}{CQ}$. I used angle bisector theorem…
user682793
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Find the area of shade region in a triangle with several small triangles inside

In the picture, the big triangle is equilateral an has area 9. The lines are parallel to the sides and divide the sides into three equal parts. What is the area of the shaded part? This is a question from a math competition I'm getting the answer…
Mers Tho
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Spring triangle problem

For an arbitrary triangle A₁B₁C₁ - what is the nearest equilateral triangle A₂B₂C₂ which has the same centroid O₁ and known side length l. Where "nearest" means minimal |A₁A₂| + |B₁B₂| + |C₁C₂| (sum of orange line lengths) Original formulation:…
Vilius Normantas
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a question about triangle

Let $(x_1,y_1),(x_2,y_2),(x_2,y_2)$ are the vertices of the triangle T. I want to show that the line $L(\alpha_3)$ defined by $$x=(1-\alpha_3-\alpha)x_1+\alpha x_2+\alpha_3 x_3$$ $$y=(1-\alpha_3-\alpha)y_1+\alpha y_2+\alpha_3 y_3$$ where…
Rosa
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Finding $\frac{BD}{AE}$ when given that $AB = AC$ and $\angle BAC = 120^\circ$

In triangle $ABC,$ $AB = AC$ and $\angle BAC = 120^\circ.$ If $D$ is the midpoint of $BC$ and $E$ is on $AB$ such that $DE \perp AB,$ find $\frac{BD}{AE}.$ I was thinking that $\angle BAC$ could be used with LOC or such methods here, but I am…
questionasker
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Why does a 3-4-5 triangle has 37° and 53° angles

A 3-4-5 triangle is a triangle with sides of the smallest integers. I am wondering why it forms a right triangle with 36.87° and 53.13° Do 36.87 and 53.13 relate to π or have some kind of ratio in some ways? Can we express 35 and 53 in some nicer…
wada
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Can this triangle be solved?

I am struggling with this question. I am unsure if it can be solved or if there is a mistake in it. Specifically, $\angle ADC$ looks to be $90^\circ$ but is not marked as such. So, given that $\angle ADC$ is unknown, can this be solved using the…
88_matsy
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Is there any illegitimate triangle with area x and sides A,B so that $A^2 + B^2 < 2 \sqrt{A^2B^2 - 4x^2}$

I was working on an equation to find out the length of one of the sides of triangle, given that the triangle's area, the variable $x$, and other two sides, $A$ and $B$, are known. Basically, if a triangle has an area of 6, and sides with lengths 3…
username
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Triangle and reflections

Let $ABC$ be a triangle where all angles measure less than $90$ degrees. $M$ and $N$ are 2 points on $BC$, in this order. $P$ and $Q$ are the reflections of $M$ and $N$ on $AB$, while $S$ and $R$ are the reflections of $M$ and $N$ on $AC$. I have to…
Wolfuryo
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Balance of a construction

I have a construction as shown in the picture. It is a post (CD) who is supported by another post (EF). You can move the point E on DC, but it is a right angle. My question is when the construction will fall in function $\mid CE \mid$, for a given…
BOB
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Given only the lengths of the three sides of a triangle, can we find if the triangle is impossible?

I was recently trying to implement Hero's Formula and I was surprised to find it declare that the perimeter of a triangle with sides of length 3, 4, and 9 is a complex number. This suggests that given the lengths of all three sides of a triangle, it…
J. Mini
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Square inside a right angled triangle.

If you have a square inscribed inside of a right angled triangle. Call the sides of the square $k$. The hypotenuse is $z$. How would you express either of the two other sides of the triangle in terms of $z$ and $k$? The corner of the square touches…
tomm0334
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If $a, b, c$ are in geometric progression and $\log a -\log 2b, \log 2b -\log 3c, \log 3c-\log a$ are in AP.

If $a, b, c$ are in geometric progression and $\log a -\log 2b, \log 2b -\log 3c, \log 3c-\log a$ are in AP. Find the type of the triangle, if it’s sides are $a, b, c$ From the given data $$b^2=ac$$ and $$2(\log 2b-\log 3c)=\log 3c-\log a +\log a…
Aditya
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How to prove that the excentral triangle passes through the vertices of the original triangle?

For a triangle ABC let's say I1, I2, I3 , are the three excentres opposite to angle A , B and C , respectively . Now if we join I1I2 , I2I3 , and I1I3 , how can we be sure that they will pass through vertex C , A and B respectively ? Please guide me…
Sameer Nilkhan
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