# Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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### Find which conditions must parameters $a$ and $b$ meet so there's exist an orthonormal basis

In $\mathbb{E^3}$ we have the plane $\pi:x-y+z-3=0$, the line $r:(2,0,1)+t(1,1,0),\ t\in\mathbb{R}$, and the point $P=(3,0,3)$. Which conditions must parameters $a$ and $b$ meet so there's exist an ...
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### Euclidean Geometry versus Analytic Geometry versus Affine Geometry?

What are the relationships (connections) among: Euclidean (or Plane) geometry Analytic geometry Affine geometry How do these things relate? I know that this is a very general question, so I'm ...
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### Prove angle addition holds for $\mathbb{R}^n$

Define $\theta(u,v)=\cos^{-1}(\frac{u\cdot v}{|u||v|})$ be the angle between $u,v\in \mathbb{R}^n$, where $u\cdot v$ is the standard inner product and $|x|=\sqrt{x\cdot x}$ for all $x\in \mathbb{R}^n$....
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### Existence of $1$-Lipschitz map between triangles

Consider two (Euclidean) triangles $T$ and $T'$. Let's say that $T$ majorizes $T'$ if there exists a 1-Lipschitz map that sends vertices to vertices and sides to sides (for some labeling of the ...
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### Nine point Circle right angled and tangent

I have been working on the following tasks for some time now and I do not know how to solve it. $(A, B, C)$ is a triangle and $K$ is his nine point circle. Show it: a) $(A, B, C)$ is right-angled if ...
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### Orthogonal complement to a graph of a linear map

Let $E$ and $F$ be two Euclidean/Hermitian vector spaces and $f:E\rightarrow F$ a linear map. Let $\mathcal{G}(f)<E\oplus F$ be the graph of $f$. Assume if it helps that $\{e_i\}_i$ is an ...
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### Find chord length given diameter and two other chords

Problem: I'm asked to find the length of $m$, given the following diagram. Note that $\overline{AC}$ = 1 and $\overline{CD} = 1$ and that $\overline{AB}$ is a diameter whose length is 4. Attempt: ...
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### Dihedral angle of a regular simplex in $n$ dimensions

For the regular simplex on $(n+1)$ points in $n$ dimensions, what is the dihedral angle i.e. the angle between two of the faces?
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### Partial derivative of euclidean distance

Hello dear mathematicians, I have two points in R3 between which I calculated the euclidean distance. Now I want to find the partial derivative with respect to the first and the partial derivative ...
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### A geometric problem for a quadrilateral

1) I have to calculate the area of a kite $$ABCD$$ with $$AB-CD=(\sqrt {2}+1)(\sqrt{3}+1)$$ and $$11 \angle A= \angle C.$$ 2) A second question is that if $$2 AB^2+AC^2+2 AD^2=4 BD^2$$ then there is ...
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### Simple problem in Euclidean Geometry — Find the radius of a circle

A student of mine brought the following question to my attention. I am currently not able to solve it, any help would be appreciated. It should be a simple circle theorem that I have now forgotten. ...
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### Prove that a line touches a circle [closed]

Let $I$ be the incenter of $\triangle ABC$. The circle passing through $I$ and centered at $A$ meets the circumcircle of $\triangle ABC$ at points $M$ and $N$. Prove that the line $MN$ touches the ...
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