# Tagged Questions

geometry assuming the parallel postulate of Euclid: 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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### Projecting line onto edge of ellipse?

I feel like the answer to this should be fairly simple, but I am absolutely hitting a brick wall here. I have a line, with angle $\beta$ and origin $(x_l, y_l)$. I have a rotated ellipse, with major ...
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### The concurrence of angle bisector, median, and altitude in an acute triangle

$ABC$ is an acute triangle. The angle bisector $AD$, the median $BE$ and the altitude $CF$ are concurrent. Prove that angle $A$ is more than $45$ degrees. Here $D,E,F$ are points on $BC,CA,AB$ ...
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### Difference between Euclidean space and $\mathbb R^3$

What is the difference between Euclidean space and $\mathbb R^3$? I have found in some books that they are the same, but in other references like Wikipedia, it says that a vector in $\mathbb R^3$ is ...
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### Sum of Determinants = Scalar product with normal vector?

Today I have seemingly simple question and maybe someone knows the answer without getting into messy calculations. So we have $n$ vectors $v_1,\dots,v_n\in\mathbb{R}^n$ and let us assume for the ...
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### Two overlapping squares

$ABCD$ is a square. $BEFG$ is another square drawn with the common vertex $B$ such that $E,\ F$ fall inside the square $ABCD$. Then prove that $DF^2=2\cdot AE^2$.
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### Shortest path between two points via two disks

Hallo everybody, I have the following problem regarding shortest paths in $R^2$. Suppose you are given two points $p$ and $q$ and two unit disks, as in the picture. I am looking for a path from $p$ ...
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### exercises for Euclid's Elements

Can you suggest some books with exercises related to Euclid's Elements, or to Euclidean Geometry, as an aid to an undergraduate course on Euclidean Geometry and its history? I need exercises that ...
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### Geometry question pertaining to a plane going through the skeleton of a cube

My question is as follows: a plane that has taken the shape of a pentagon is intersecting the skeleton of the cube. Or I guess we could think of it as a cross section. Points $M$ and $N$ were used to ...
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### Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
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### There exist a bijection between the Real numbers and points of a straight

Assuming that we are building our geometry on the axioms of Euclid/Hilbert, and using either the Dedeking or Cauchy construction of the reals, how can one prove this statement? I've looked up on the ...
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### Banach Tarski paradox in Minkowski space

The Banach - Tarski paradox states: 'A three-dimensional Euclidean ball is equidecomposable with two copies of itself.' My simple question is: Does this paradox hold his validity also in the Minkowski ...
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### IMO 2014 problem 3, first day

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such ...