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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V.I. Arnold says Russian students can't solve this problem, but American students can -- why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is $10$ inches, the altitude dropped onto it is 6 inches. Find the area of the triangle. American…
Eli Rose
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Is the blue area greater than the red area?

Problem: A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is red. Which area is greater? Let the area of each…
Mr Pie
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What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

My friend gave me this puzzle: What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges? I tried to draw the picture and I drew a smaller (concentric)…
terrace
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Different proofs of the Pythagorean theorem?

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's are some of the most elegant proofs? My favorite is this graphical one: According to…
Shane
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A new pythagorean proof

I am writing to you today because I believe I may have developed a new visual proof for the Pythagorean theorem. The proof is based on a geometric dissection method, equating the area of a larger composite square to the sum of its smaller…
kingyoon
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What's a proof that the angles of a triangle add up to 180°?

Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point: However, now that I'm in university, I'm not convinced that this proof is the best one (although…
Joe Z.
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Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates given and then apply Heron's formula. Is this the…
TSP1993
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How to calculate the area of a 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. (119.91227722167969, 122.7717056274414, 39.3568115234375),…
iamgopal
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Do two right triangles with the same length hypotenuse have the same area?

I watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with the same length hypotenuse have the same area. The…
totymedli
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What is the name of this theorem of Jakob Steiner's, and why is it true?

In The Secrets of Triangles a remarkable theorem is attributed to Jakob Steiner. Each side of a triangle is cut into two segments by an altitude. Build squares on each of those segments, and the alternating squares sum to each other. The book…
MBP
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Is being a right triangle both necessary and sufficient for the Pythagorean Theorem to hold?

I recently encountered a Stack Overflow question (since closed) in which the OP was testing for whether a triangle was right by whether or not it "met" the criteria of the Pythagorean Theorem (i.e. whether or not the square of the hypotenuse is…
EJoshuaS - Stand with Ukraine
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Crazy fact(?) about circles drawn on base of triangle between cevians: they always fit, no matter what their order?

Take any triangle, and draw any number of cevians from the top vertex to the base, with any spacing between the cevians. In each sub-triangle thus formed, inscribe a circle. Now rearrange the order of the circles from left to right (but don't…
Dan
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Probability that 3 points in a plane form a triangle

This question was asked in a test and I got it right. The answer key gives $\frac12$. Problem: If 3 distinct points are chosen on a plane, find the probability that they form a triangle. Attempt 1: The 3rd point will either be collinear or…
Serenity
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If $(a,b,c)$ are the sides of a triangle, what is the probability that $ac>b^2$?

Let $a \le b \le c$ be the sides of a triangle inscribed inside a fixed circle such that the vertices of the triangle are distributed uniformly on the circumference. Question 1: Is it true that the probability that $ac > b^2$ is $\displaystyle…
Nilotpal Sinha
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Triangle inequality for subtraction?

Why is $|a - b| \geq|a| - |b|$?
CodeKingPlusPlus
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