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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26 views

How many points $P$ such that $\angle APB=\angle BPC=\angle CPA $ are there?

Given that $\triangle ABC$ is arbitrary. How many points $P$ such that $\angle APB=\angle BPC=\angle CPA $ are there?
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1answer
61 views

Why ternary diagrams work

I am trying to understand why ternary diagrams work. In order that the altitude criterion be valid, if I correctly understand, given equilateral triangle $ABC$, whose vertices I name as the three ...
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1answer
43 views

“Polysticks” in 3d

Consider a finite set of three-dimensional Euclidean vectors with integer components. How many three-dimensional closed loops can I construct with them? How many of them are elementary, i.e., cannot ...
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1answer
121 views

Volume of the intersection of two tetrahedra

First, I am far from a mathematician, and this question may be easy, if that's the case, please don't hesitate to let me know. Suppose I have 2 tetrahedra (2 3D simplex), with known ...
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1answer
165 views

euclidean geometry books…

I consider myself poor in plane euclidean geometry. so I need a good geometry book which contains very good theory, and a collection a large number of solved problems, and the end of each part.This ...
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1answer
101 views

Is there a name for the recursive incenter of the contact triangle?

Recently, I became aware that there are many more triangle centers than the four I learned about in school. This reminded me of a thought I had when I first learned about the incenter: what point ...
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12 views

Dual Objects and Symmetries

In the study of symmetries of platonic solids (tetrahedron, cube, octahedron, ..), I came across the following. Group of rotational symmetries of a cube is $S_4$. Since octahedron is dual to cube, ...
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23 views

Mathematics-Oriented 4-D Glossary?

Is there somewhere a comprehensive glossary of words or phrases describing geometric concepts or objects in the Euclidean (not Einsteinian) fourth dimension? I have seen a glossary which purported to ...
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1answer
31 views

Euclidean problem of geometry

Let the two quadrilaterals ABCD and EFGH been given: Let's take these hypothesis: $AD = EH$ $A\hat{B}D=A\hat{C}D=E\hat{F}H=E\hat{G}H$ $AC=EG$ The triangle ABD is isosceles and equal to the ...
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9 views

Intersection/union of hyperballs in Minkowski space

I'm trying to manipulate hyperballs bounded by the Minkowski distance. What I would like to do is take the intersection/union of two hyperballs and then find the smallest hyperball which covers the ...
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1answer
62 views

calculate points coordinates on plane from their distances matrix

Given a list of points on a plane is simple to generate a distances list between each pair of points. Pseudo Code: ...
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3answers
273 views

Scaling and rotating a square so that it is inscribed in the original square

I have a square with a side length of 100 cm. I then want to rotate a square clockwise by ten degrees so that it is scaled and contained inside the existing square. The image below is what I'm ...
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1answer
32 views

What is the fundamental theorem on discrete groups of Euclidean spaces?

I have been reading the book Using Algebraic Geometry by David A. Cox, John Little, Donal O'Shea for a university project. I am not clear as to what exactly in meant by the phrase "the fundamental ...
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3answers
73 views

Every reflection is an isometry proof

The theorem is that every reflection $R_{S}$ in an affine subspace $S$ of $\mathbb{E}^{n}$ is an isometry: $R_S:\ \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}:\ x \mapsto R_{S}(x) = x + 2 ...
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1answer
75 views

At most $2n$ vectors, the angle between which $\geq\pi/2$.

In a previous question it is proved that in $\mathbb R^n$ there are at most $n+1$ vectors, the angle between which $>π/2$. How to prove that there are at most $2n$ vectors, the angle between which ...
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1answer
50 views

Alternative word for Euclidean Geometry

If Euclid has only collected the geometry stuffs while books of the other geometer have been burnt, calling the main branch of geometry under name of him might look academically unethical for some ...
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32 views

If three circles have two common points, prove that every circle that is orthogonal to two circles is also orthogonal to third.

Three circles are given $k_1$,$k_2$,$k_3$ that have two common points A and B. Prove that every circle $k$ that is orthogonal to circles $k_1$,$k_2$, is also orthogonal to $k_3$. Here is my proof ...
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1answer
75 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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1answer
76 views

At most $n+1$ vectors, the angle between which $>\pi/2$.

In a $n$ dimensional Euclidean space $V$, there exists at most $n+1$ vectors, each pair has inner product $<0$. This is geometrically obvious in $3$ dimensions...But how can we prove it ...
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30 views

On the congruence of triangles

Sorry for the perhaps somewhat trivial question, but are the criteria for the congruence of two triangles, i.e. "side-angle-side", "side-side-side" and "angle-side-angle", taken as postulates or can ...
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1answer
135 views

Solve $10x+2x^2+x^3=20$ using only algebra and geometry?

The cubic formula and modern math is not allowed, only algebra, geometry, and the like. Supposedly this problem was given to Fibonacci. Here is the whole paragraph I read: In Flos Fibonacci gives ...
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1answer
15 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
42 views

Constructing the inverse of a number geometrically.

this picture: shows a way to construct the inverse of a number $a\ge1$. but how can we construct for a number that is less than 1? My try:: Q1: is my try correct? Q2: how to prove ...
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54 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
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93 views

Why are these three lines concurrent?

Consider a triangle $ABC$ with incentre $I$ and let $AI \cap BC=D$. Let the incentres of $\triangle ACD$ and $\triangle ABD$ be $E$ and $F$ respectively. Prove that $AD$, $BE$ and $CF$ are ...
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506 views

If $d$ is a metric, is $d(x,y)/(1+d(x,0)+d(y,0))$ a metric?

I now that one can show that if $d$ is a metric on a vectorspace $X$ then so is $$\varrho(x,y):=\frac{d(x,y)}{1+d(x,y)}.$$ This easily follows from the fact that the function $s \mapsto \frac{s}{1+s}$ ...
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2answers
207 views

Drawing Euclid?

I decided to study Euclid for fun. I have Oliver Bryne's edition. I also want, as much as possible, to construct the figures myself, to get a deeper understanding. How did people traditionally do ...
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1answer
21 views

Comparison of the Chebyshev centers and radii of a set and of its bounding box

The Chebyshev center of a bounded set $Q$ having non-empty interior is defined in this question as the center of the minimal-radius ball enclosing the entire set $Q$. Let $B$ be the minimum-volume ...
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1answer
25 views

Rotation of curve function

I am working on some coding where I require expertise in field of Mathematics. I have a function: $$F(x) = -0.007x^4 + 1.971x^3 - 190.4x^2 + 8150x - 13024$$ I want to rotate some section of this ...
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0answers
34 views

Estimating distance between two points in n-dimensional space, with knowledge of other paths

Suppose there exist four randomly distributed points in $n$-dimensional space: $A$, $B$, $C$, and $D$. We have no knowledge of the coordinates of any of these points, but we do know nearly all of the ...
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1answer
63 views

What's the exact definition of symmetry?

My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this: Let $X \subseteq R^2$. A symmetry of $X$ is an ...
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56 views

Can Point inside quad be determined with angles alone?

Given A Quad($C$, $D$, $E$, $B$) and Points $A$, $G$, $F$ Question Is it possible by calculating the angles between points to determine whether a point is inside (including on), or outside the quad ...
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1answer
23 views

Inner product respect on a non-canonical base

Let a,b be vectors, on the standard base we use the dot product by simply doing a.b. But when we consider an other base we put a symmetric matrix between them. Why? How does that work? Thanks
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1answer
13 views

Does this operation result in a convex set

Denote by $cm(B)$ the center of mass of the set $B\subseteq\mathbb{R}^2$. Given two convex sets $A,X\in \mathbb{R}^2$, define $Y$ in such a way that $X\cap A\neq \emptyset$ if and only if $cm(A)\in ...
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1answer
88 views

Area of a 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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1answer
66 views

who discovered the orthocenter of a triangle?

I tried to answer Is there a name for this result in planar geometry? and wanted to go back to the first mention of the orthocenter (or even the altitude of a triangle, but i did draw a complete ...
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1answer
56 views

Is there a name for this result in planar geometry?

I found out that the following statement is fairly easy to prove: Let $A$, $B$ and $C$ be thee distinct points in the plane. Let $S_{AB}$ be the circle that has the line segment $AB$ as a ...
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4answers
161 views

What happens to the angles of an isosceles triangle if one vertex is at infinity?

My son and I were trying to decide whether an isosceles triangle can ever have 90 degree base angles. I would argue that if the two equal length sides are both infinitely long, they must have 90 ...
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1answer
29 views

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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4answers
113 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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44 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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0answers
57 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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46 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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1answer
27 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) ...
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1answer
67 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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45 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
64 views

Euclidean geometry and irrational numbers.

I was wondering, given a square that is $1 \times 1$, how can we know that the diagonal is an irrational length geometrically??? We could use the Pythagorean Theorem to see that the diagonal of a ...
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1answer
33 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
82 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
28 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 ...