# 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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### Are the only quadrilaterals satisfying this symmetric relation rectangles?

$\newcommand{\S}{\mathbb{S}^1}$ $\newcommand{\la}{\lambda}$ While solving an optimization problem, I reached the following question: Let $x_1,x_2,x_3,x_4 \in \S$ be four distinct points on the unit ...
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### Find the length of CX

I found this question on Twitter My Attempt: I marked the circumcenter of the circle as O and drew radii $OA, OB, OC$ and with some angle chasing, I found the angles marked in the diagram attached ...
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### $k$-dimensional subspace in $\mathbb R^n$ such that the orthogonal projection of standard basis $e_1, \dots, e_n$ have the same length.

I wonder how to construct or at least prove the existence of a $k$-dimensional subspace $V_k$ in $\mathbb R^n$ such that the orthogonal projection of canonical basis $e_1, \dots, e_n$ onto $V_k$ have ...
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### On the possible cardinalities of Sylvester lines of sets of points in the real plane.

Let $n$ be a natural number greater than or equal to $3$. Also, let $S$ be a set of $n$ points in the real plane $\mathbb{R}^2$, such that there is no line that passes through all points of $S$. I ...
### The upper bound for the radius of the $k$-dimensional balls contained in an $n$ dimensional unit hypercube can be attained
It is shown in this thread that $\dfrac{1}{2}\sqrt{\dfrac{n}{k}}$ is an upper bound for the radius of $k$-dimensional balls that can be contained in an $n$ dimensional unit hypercube. But I have ...