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Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

The geometry of Euclid is based on five axioms (Euclid called them postulates). Any geometry based on the first four of these is called an absolute geometry. The fifth one states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

It was observed by Proclus that, in the presence of the the other four postulates, Euclid's fifth postulate can be replaced by Playfair's axiom:

Given a line and a point not on it, then one and only one line parallel to the given line can be drawn through the point.

The independence of the parallel postulate and its equivalent formulations from the first four axioms was shown by Beltrami in 1868.

Another alternative definition is that two lines are parallel if every perpendicular extended from one meets the other as a perpendicular.

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Why doesn't the manhattan distance tend towards the euclidean distance as the number of subdivisions become infinite?

Suppose that we have a unit square and are interested in the distance between two opposite corners. The euclidean distance is $\sqrt{2}$. The manhattan distance is $1 + 1 = 2$. Suppose we subdivide the square by divide both the width and height in…
Cedric Martens
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Points in $R^3$ that lie in an octahedron (finding a frame that minimizes the L1 norm for a set of points)

Four points $\vec{a}_i$ in $R^3$ are given such that $\vec{a}_1+\vec{a}_2+\vec{a}_3+\vec{a}_4=\vec{0}$ and $|\vec{a}_1|+|\vec{a}_2|+|\vec{a}_3|+|\vec{a}_4|\leq 2$. I want to find the smallest octahedron such that the following 16 points lie inside…
user1105749
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What is the intersection of $2$ hyperspheres?

Given $a ,b \in \Bbb R^+$ and ${\bf x}, {\bf y} \in \Bbb R^n$, where ${\bf x} \neq {\bf y}$, how can I characterize the following set? Is it a hyperplane? $$ \left\{ {\bf c}_1 \in \Bbb R^n : \| {\bf x} - {\bf c}_1 \|_2 = a \right\} \cap \left\{ {\bf…
entropy
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Is there a quick method to find the equation of a parabola?

I solved this question but it is very lengthy. Question: Find the equation of parabola with vertex $(2,-3)$ and focus $(0,5)$. I'm adding here my answer which is lengthy enter image description here
Raj Agarwal
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How to find the measure of BC in this triangle?

In this right triangle, AB = 1, BD= 1/2, and AD is a bisector. How do I find CD? I tried finding AD with pythagora's theorem. Which would be square root of 3 over 2. Which should be equal to CD: But it seems my logic is wrong. Can anyone help me?
user3347814
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How to place 4 identical non-overlapping rectangles with corners on 4 randomly preplaced points?

2D Euclidean geometry. I'm not sure if this problem was already asked I tried to search but I failed to find similar. So imagine we have 4 randomly placed points on a plane. And we also have 4 identical rectangles that are "parallel" - i.e. their…
Harmonic Kolobok
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Elementary proof of d(circular segment area)/ d(circular segment length) = radius?

It is easy to use calculus to find that in a circular segment, if the chord is held constant, then $$\left(\frac{\partial a}{\partial s}\right)_c = R$$ Where $c$ is the chord, held constant, $a$ is area of circular segment and $s$ is arc length. Is…
MaudPieTheRocktorate
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Construct quadrilateral given the length of its four sides plus the length of the segment joining two opposite sides

Good evening to everybody. I was reading today a chapter on a Euclidean geometry book concerning quadrilateral constructions and there was an exercise about how to construct a quadrilateral ABCD if we are given the length of its four sides plus the…
Petros Karajan
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Construct quadrilateral given the length of its diagonals and the angle between them and two opposite angles

Good afternoon to everybody. Today I was reading a chapter on an Euclidean geometry book related to the theorem of Ptolemys and the theorem of Brahmagupta and there was an exercise about how to construct a quadrilateral(not necessary inscribable) if…
Petros Karajan
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Euclid Geometry problem

I'm trying to solve this problem, I think I almost made it, at least I hope. I don't know where I'm wrong. Let I be the incenter of a triangle $ABC$ ($AB < AC$). The line $AI$ intersects the circumcircle of $ABC$ again at $D$. The circumcircle of…
Math
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Euclidean geometry problem without trigonometry

$ABC$ is a triangle such that the median divides the angle $A$ in $15$ e $30$ degress. What the measure of the other two angles? I tried building a parallelogram on the MC side but I didn't know nothing of useful. The answer is $30$ degrees one and…
Math
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A plane geometry tough problem

$ABCD$ is a quadrilateral. $P,Q,R,S$ are the midpoints of $AB,BC,CD,DA$ respectively. $PR$ and $SQ$ intersect at $L$. $T$ is any point within the quadrilateral. Prove that $4LT^2+LA^2+LB^2+LC^2+LD^2=TA^2+TB^2+TC^2+TD^2$.
sangabattulalokesh
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I want to rotate a point on a sphere surface

I want to rotate a point on a sphere surface . I was instructed as I can use Rodrigues rotation formula , (I thank ja72 very much). I tried to use the formula but it did not work . I can not find where I was wrong . I wrote a short Scilab program…
seven_swodniw
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Radial distance for superellipsoid

Consider a point $[x_0\,\,y_0\,\,z_0]'\in\mathbb{R}^3$ and define the inside-outside function \begin{equation*}F(x_0,y_0,z_0)\triangleq…
matteogost
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A criterion in finding a line segment parallel to a pair of parallel chords in a circle

I would appreciate either an explanation or a citation to the following statement in Euclidean geometry. $\overline{\mathit{AB}}$ and $\overline{\mathit{CD}}$ are chords of a circle such that $\overline{\mathit{AD}}$ and $\overline{\mathit{BC}}$ are…
user74973
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