Questions tagged [triangles]
For questions about properties and applications of triangles.
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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. ...
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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:...
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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 ...
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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 ...
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uniform random point in triangle in 3D
Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. This paper (section 4.2) says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following ...
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