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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Is there a formula for finding the centers of the faces of a platonic solid?

Is there a formula for finding the centers of the faces of a platonic solid given the center of the first (origin) face to be $P_0(x_0,y_0,z_0)$?
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26 views

Area of weird circumcenter triangle equals area of medial triangle

Let $X$, $Y$, $Z$ be the midpoints of sides $BC$, $AC$, $AB$ respectively in triangle $ABC$. Let $O_{A}$, $O_{B}$, and $O_{C}$ be the circumcenters of triangles $AZX$, $BXY$, and $CYZ$ respectively. ...
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42 views

A question concerning radians and arc length

I was asked by a colleague yesterday about how the formula for the arc length of a circle is derived. I wanted to give them a correct answer, so I said I'd get back to them once I'd thought about it ...
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27 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
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13 views

How to determine roll-pitch-yaw parameters from homogeneous transformation matrix

We have a transformation matrix $T = \begin{pmatrix} cos(\theta_1) & sin(\theta_1) & 0 & l_1 cos(\theta_1) \\ sin(\theta_1) & -cos(\theta_1) & 0 & l_1 cos(\theta_1) \\ 0 ...
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74 views

Problem of axiomatic euclidean geometry

Let the usual five postulates of Euclid been given. Let's take also this postulate: "If two points lies on the same plane, the whole straight line joining the two points lies on that plane". Is it ...
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27 views

Intesection point of feet of altitudes

If triangle has vertexes at $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, is the intersection points of feet of altitudes $$x_h = \frac{x_1x_2(y_2-y_1) + x_2x_3(y_3-y_2) + x_3x_1(y_1-y_3) + y_1^2(y_3-y_2) + ...
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1answer
23 views

circumscribing sphere of tetrahedron

What are the conditions under which the center of circumscribing sphere of a tetrahedron is located inside(outside, face, edge) of the tetrahedron? In other words, how can we define acute(obtuse) ...
3
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1answer
30 views

Prerequisites for Hartshorne: Euclid and beyond?

as the title suggests, I am looking for the prerequisites to Hartshorne's Euclid and beyond. I just found this book and I think it's wonderful, but the downside is that I only know math up to single ...
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26 views

How to smooth a list of angles.

I'm not a math guy so maybe there is a super simple thing that my eyes cannot see. And sorry if my math terminology is not good at all. Please address me the right math terminology to use because ...
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44 views

Pentagon Forms a 10-sided Polygon Ratio Problem

Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length $1$. The sides of the pentagon are extended to form the $10$-sided polygon shown in bold in the picture that I have attached. Find the ...
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18 views

On Euclid's definition of similar and equal solid figures.

The Euclid's definition of similar solid figures is Similar solid figures are those contained by similar planes equal in multitude. And the Euclid's definition of equal solid figures is Equal and ...
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1answer
29 views

Euclidean isometries

I am asked to show that every translation of the euclidean plane can be written as two reflections. How do I proceed (algebraicly)? My idea is to proof it in a sense of creating a rectangular ...
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3answers
67 views

What's the best polygon for tiling the plane?

We want to cover the whole plane by tiles, shaped as a polygon with equal-length sides, such that there is not overlapping and any gap (Note that all the tiles are similar to each other). which ...
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1answer
25 views

Proving elementary property about hyperplanes.

I am currently working through a textbook, and I am having some problem with the following problem: Define a hyperplane to be an $(n-1)$-plane of $E^n$. Prove that $P$ is a hyperplane if and only ...
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1answer
21 views

Understand to paragraphs in Chicones book concerning tangent vector fields

I apologize in advance for the length of this question.... Now, consider the sphere $$ S:=\{ (x,y,z):x^2+y^2+z^2=1 \}$$ in $\mathbb{R}^3$. Now in Chicone's book "Ordinary differential equations" on ...
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22 views

$1,2,3$ task — calculate tangents

Given is square $ABCD$. Point $E$ is the midpoint of segment $CD$ ($E\in CD \wedge |DE|=|EC|$). Point $P$ is common point of diagonal $AC$ and line segment $BE$. ($\lbrace P\rbrace = AC \cap BE$). ...
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1answer
17 views

Prove that a planar bipartite graph on n nodes has at most 2n−4 edges.

I know that we have to use Euler's formula ( v−e+f=2) but I don't understand how f = e/2.
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1answer
27 views

How to find the equation of a hyperplane in $\mathbb R^4$ that contains $3$ given vectors

Find the equation of a hyperplane in $\mathbb R^4$ that contains $3$ given vectors. The vectors are: $$v_1 = (1,0,-1,0),\quad v_2 = (0,1,1,1),\quad v_3 = (1,1,-1,0) $$ I've found the equation ...
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1answer
35 views

An interior point in the triangle

Suppose $P$ is an interior point of a triangle $ABC$ and $[AP]$, $[BP]$, $[CP]$ meet the opposite sides $[BC]$, $[CA]$, $[AB]$ in $D$, $E$, $F$ respectively. Find the set of all possible values of the ...
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1answer
37 views

Euclid I.24 Proof Why is DFG greater than EGF?

Proposition 24 If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the ...
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1answer
39 views

divide a triangle into two parts equivalent through the point P [closed]

Given a point $P$ on the side $AB$ of a triangle $ABC$. Draw a line through $P$ that divides this triangle into two equal areas.
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1answer
61 views

Proof involving chords of a circle

In a circumference with center $O$, three chords $\overline{AB},\overline{AD}$ and $\overline{CB}$ such that the last two intersect in $E$. Show that $AE·AD+BE·BC=AB^2 $. Added: ...
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4answers
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Does a triangle always have a point where each side subtends equal 120° angles?

Is there a point $O$ inside a triangle $\triangle ABC$ (any triangle) such that the angle $\angle{AOB} = \angle{BOC} = \angle{AOC}$? What do we call this point?
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Thales theorem practical application

I wonder if someone know some interesting/practical applications of Thales theorem in triangles. (but not using the similarity of triangles) Thanks a lot!
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64 views

Which of these are constructible numbers?

Which of these are constructible numbers? $$-\frac35\quad,\quad27^\frac16+2i\quad,\quad2^\frac13\quad,\quad e^{\frac{\pi i}{10}}$$ Please tell me how you came to the answer too. Thanks!
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4answers
83 views

Geometry question involving triangles given with picture.

Here's the question: $\overset{\Delta}{ABC}$ is a triangle. $D$ is a point on $[BC]$. $|BD|=4$. $|AD|=|CD|$. $\text m(\widehat{CBA})=\alpha=30^\circ$. $\text ...
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11 views

Find the area of the region formed at the intersection of all these arcs (GEFH) in terms of a.

ABCD is a square and the arcs centered at the vertices of the square and the radii, are all equal to the side-lengths of the square, (=a). I feel like the arc lengths should all measure 90 degrees ...
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1answer
24 views

Find the measure of the angle ADE and the radius of the circles in terms of the sides of the square ABCD.

We know that the triangles are congruent. The circles are congruent. And EFGH is a square. I don't even know where to begin with this one besides to say that angle ADE is equal to 90-m(EAD), but I ...
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1answer
23 views

Condition for dim of the Euclidean space with orthogonal basis

I would like to show that if the orthogonal basis of the $\Bbb R^n$ Euclidean space with the standard dot product has the vectors whose elements are exclusively $1$ or $-1$, then $n \le 2$ or $n$ is ...
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2answers
37 views

Questions about elipse

Given the center of an elipse and three of its points, is this elipse completely determined? What is the easiest way to show that five points of an elipse are enough to determine the elipse?
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1answer
143 views

How can we draw a line?

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this?
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1answer
22 views

Specific triangle, symmetral of angle proof

In triangle $ABC$, angle $\gamma = 120$. Prove that $|\overline{CC'}|=\tfrac{ab}{a+b}$, where $\overline{CC'}$ is symmetral of angle $\gamma$ inside triangle. Look at image. I can't use areas, ...
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1answer
32 views

How to represent two coordinate system transformations as one

I'm working on a system of relative euclidean coordinate systems. I'd like to define every coordinate system relative to a global coordinate system, which I'll refer to as [0]. Then, for example, ...
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1answer
370 views

The number of the circles which are tangent to two circles and to a line

Suppose that we have two distinct circles and a line on a plane and that the distance between the centers of the circles is bigger than the sum of their radiuses. Also, suppose that the two circles ...
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0answers
15 views

Dandelin spheres and the asymptotes of a hyperbola

The other day, I was reading up on the synthetic geometry of conic sections a bit, and I wondered: is it possible to construct the hyperbola's asymptotes given just the intersecting plane and the ...
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16 views

What is the analog of the scalar triple product in two dimensions?

Is there a standard name and/or a notation for the analog of the scalar triple product in two dimensions? Namely, i am interested in the following operation: given two elements $\vec u$ and $\vec v$ ...
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23 views

Circumcentre of three points X, Y, Z, given distance from each to points A and B

I'm racking my brain trying to figure out where to start on this, and it's been too many years since working on these kinds of problems. I have six measurements which I'd like to use to calculate a ...
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6 views

What is the proper name of a point a long a smooth curve where the radius changes but not direction of curvature?

What you call a point a long a smooth curve where the radius changes? When it reverses curvature, it’s an “inflection point”. What if it doesn’t change direction, just radius? I seem to remember ...
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1answer
46 views

euclidean distance between one dimension points-how to? [closed]

i am reading a research paper about round-robin scheduling algorithm that uses Euclidean distance to determine a time quantum based on similarity of burst times of all processes in the ready queue,but ...
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2answers
68 views

How long is the curve that a creature walks?

I have a problem in solving mathematical problem. Take a ball with radius 60 cm. A creature walk from the southpole to northpole by following the spiral curve that goes once around the ball every ...
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3answers
52 views

5 points on a plane with rational distances

Can you find 5 points on a plane whose Euclidean distances between them are all rational numbers and no 3 points out of them are co-linear? If the answer is yes, can we find a construction for ...
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1answer
17 views

Tranversal parallel lines theorem

I need to prove Tranversal parallel lines theorem that says: If two parallel lines are cut by a transversal, the corresponding angles are congruent, the alternate angles are congruent, and the ...
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54 views

Calculating Euclidean dissimilarity for a given cluster with itself

Suppose I have clusters $$A= \{(1,1)^T, (1,2)^T\} $$ $$B=\{(2,3)^T, (3,4)^T\} $$ $$C= \{(4,5)^T, (5,6)^T, (1,2)^T\} $$ I wish to use the Euclidean dissimilarity and Average linkage to calculate a ...
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1answer
26 views

A triangular inequality including squares of sides

Show that for any triangle ABC, the following inequality is true $$a^2 + b^2 + c^2 > \sqrt{3} \max\{|a^2-b^2|,|b^2-c^2|,|c^2-a^2|\}$$ where $a,b,c$ are the sides of the triangle
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19 views

Central symmetry

There's definition: Central symmetry $s_O:M \to M$ is bijection defined as $s_O(T)=T'$ if and only if $O$ is midpoint of $\overline{TT'}$. 1st: Prove that central symmetry is involution ($s_O \circ ...
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1answer
224 views

Proof of a certain lemma in geometry

In the following article: http://yufeizhao.com/olympiad/geolemmas.pdf in the proof of the fact about the diameter of the incircle on page 2, the author claims that the proof that $BD = CF$ follows ...
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1answer
22 views

How can I uniformly draw points from an ellipsoid?

Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the ...
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1answer
23 views

Pairs of isometries that jointly fix a set (revised)

Let $E$ be a Euclidean space, and let $X, Y\subseteq E$ be two disjoint subsets. Suppose that there exist isometries $i$ and $j$ such that $i(X)\cap j(Y)=\emptyset$ and $i(X)\cup j(Y)=X\cup Y$. Does ...
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Inequalities in a quadrilateral

In a quadrilateral ABCD, it is given that AB is parallel to CD and the diagonals AC and BD are perpendicular to each other. Show that ...