Questions tagged [euclidean-geometry]
For questions on geometry assuming Euclid's parallel postulate.
1,205 questions
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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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Compass-and-straightedge construction of the square root of a given line?
Given
A straight line of arbitrary length
The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude ...
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How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?
I would like to find the apothem of a regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)?
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Geometry Book Recommendation?
Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...
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Showing that an Isometry on the Euclidean Plane fixing the origin is Linear
Suppose $f$ is an isometric (i.e., distance preserving) function on $\mathbb{E}^2$ such that $f(0,0) = (0,0)$. Then I want to show that $f$ is necessarily linear. Now $f$ is linear iff $f$ is both ...
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Looking for an alternative proof of the angle difference expansion
I have thought about this for a while and have no progress.
Does there exist a purely Euclidean Geometric proof of the Angle Difference expansion for Sine and Cosine, for Obtuse angles?
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Book recommendation on plane Euclidean geometry
I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
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Formal Proof that area of a rectangle is $ab$
I tried to prove that the area of a rectangle is $ab$ given side lengths $a$ and $b$.
The best I can do is the assume the area of a $1\times1$ square is $1$. Then not the number of $1\times1$ squares ...
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Determining if an arbitrary point lies inside a triangle defined by three points?
Is there an algorithm available to determine if a point P lies inside a triangle ABC defined as three points A, B, and C? (The three line segments of the triangle can be determined as well as the ...
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What is the modern axiomatization of (Euclidean) plane geometry?
I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
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Euler angles and gimbal lock
Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
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Why do we use the Euclidean metric on $\mathbb{R}^2$?
On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used:
$\pi$ is the area of the unit circle.
But what is a circle?
A circle is the set of tuples $(...
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Show that the angles satisfy $x+y=z$
How can I show that $x+y=z$ in the figure without using trigonometry? I have tried to solve it with analytic geometry, but it doesn't work out for me.
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Why is Euclidean geometry scale-invariant?
In Euclidean geometry, I frequently use concepts related to invariance under scaling. For example, I know that if two squares have different side lengths, the ratio of their side lengths is the ...
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A conjecture related to a circle intrinsically bound to any triangle
Given a triangle $ABC$, whose (one of the) longest side is $AC$, consider the two circles with centers in $A$ and $C$ passing by $B$.
(The part in italic is edited after clever observations pointed ...
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