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.
0
votes
1answer
17 views
Problem on hyperbolic hyperboloid generated by a rotation
This is the problem:
In $\mathbb{E}^3$ we consider the conic $\gamma$ of equations $x=yz-2=0$ , the line $a$ of equations $x=y+z=0$ and the surface $Q$, that is generated by the rotation of $\gamma$ ...
4
votes
0answers
46 views
A numerical coincidence with continued fractions
My brother built a garage that measures 45 feet by 30 feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches,
$$
...
3
votes
1answer
51 views
What accounts for the special relationship between Euclidean geometry and other branches of math?
Many times there are problems which are, in a sense "outside of geometry", but are nevertheless amenable to a geometric approach. For example, I may be asked to prove that the ranges of any two affine ...
3
votes
1answer
48 views
Determining the embedding space:
I have seen a lot of discussion of alternate geometries for example on a sphere or hyperbolic saddle as opposed to a plane:
Has anyone consider the notion of that plane or hyperbolic saddle itself ...
0
votes
0answers
41 views
How to prove the property of scalar distribution over vector addition when the vectors are collinear?
$\overrightarrow{a},\overrightarrow{b} \in V^3 , \alpha \in \mathbb{R} $
Prove: $\alpha(\overrightarrow{a} +\overrightarrow{b}) = \alpha\overrightarrow{a} + \alpha\overrightarrow{b}$
When $\alpha = ...
2
votes
1answer
35 views
Any name for an isosceles triangle sides
Is there an English translation for Finnish words kanta and kylki? Namely, if $ABC$ is an isosceles triangle with $AB=AC$ then $BC$ is kanta in Finnish and $AB$, $BC$ are both kylki.
1
vote
1answer
25 views
Complex 3-D Euclidean space - inner product
1st question: Lets say we have a 3-D complex euclidean space. How do we geometrically draw this space? if 3-D real Euclidean space is represented by these base vectors:
2nd question:
Is there a ...
1
vote
1answer
42 views
An explicit bijection between $R^n$ and $S^n\setminus \{$point$\}$
I am trying to formulate a bijective map $f_n:S^n\setminus (1,0,\dots,0)\to R^n$. I consider $S^n$ in $n$-spherical coordinates, that is, $S^n = \{(r,\phi_1,\dots,\phi_n)\in R^{n+1}\ |\ r = 1\}$, and ...
4
votes
2answers
69 views
Fit a equilateral triangle on three arbitrary parallel lines with an edge and compass
How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?
1
vote
1answer
50 views
Meaning and types of geometry
I heard that there's several kind of geometries for instance projective geometry and non euclidean geometry besides the euclidean geometry. So the question is what do you mean by a geometry, do you ...
4
votes
1answer
42 views
Optimal rotation to align a circle with external points
I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
4
votes
1answer
63 views
Angular alignment of points on two concentric circles
I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
3
votes
1answer
87 views
Finding intersection of 2 planes without cartesian equations?
The planes $\pi_1$ and $\pi_2$ have vector equations:
$$\pi_1: r=\lambda_1(i+j-k)+\mu_1(2i-j+k)$$
$$\pi_2: r=\lambda_2(i+2j+k)+\mu_2(3i+j-k)$$
$i.$ The line $l$ passes through the point with ...
23
votes
1answer
495 views
About Euclid's Elements and modern video games
I just watched this video about Euclid's treatise the Elements. I got introduced to the postulates and a couple of propositions of book I. I really liked this video, I'm not sure if this is because of ...
1
vote
2answers
89 views
Finding the locus of the midpoint of chord that subtends a right angle at $(\alpha,\beta)$
There is a circle $x^2+y^2=a^2$. On any line that cuts the circle in two distinct points(it is a secant), the points of intersection with circle are taken and at those two points I draw the tangents ...
0
votes
1answer
27 views
Construction of an isosceles trapezoid given 3 different lengths.
Given 3 different lengths, how do you construct an isosceles trapezoid when two of these lengths are bases and the other a side.
1
vote
2answers
60 views
Approximating Euclidean geometry, restricted to $\mathbb{Q}$
I'm having trouble putting this into a fully coherent question, so I'll give the broad question, then a few bullet points to give you a better idea of what I'm asking.
I'm looking for a line of ...
3
votes
4answers
130 views
Maximum surface inside a triangle
If I have a triangle with sides of length a, b, c and I have a rope of length L,
what is the maximum surface of a boundary I can form with that rope that is entirely
inside the triangle.
Normally, ...
3
votes
1answer
50 views
maximising the angle $\theta$
OK, suppose I have two points in cartesian coordinate system, say $P(x_1,y_1)$ and $Q(x_2,y_2)$. I have a line as well, that is, for simplicity
$$y=mx$$
Assuming that
$$y_1\neq mx_1,y_2\neq mx_2$$
I ...
0
votes
1answer
79 views
What's the logical flaw in Euclid's construction of the triangle?
NJ Wildberger says in this video that there's a logical flaw in Euclid's construction of the triangle, that you're not really able to know (apart from the picture) if the circles intersect. He also ...
5
votes
1answer
34 views
Sum of medians of a triangle
I'm very confused because I don't know how I can prove that the sum of the medians of a triangle is equal to the vector zero. Can someone give me a tip or something? Thanks! (And sorry if this ...
0
votes
0answers
31 views
Distance to a convex polyhedron: about different approaches
I know there are a lot of litterature out there about convex polyhedra and distance computation, but I don't quite catch which one has the best computational complexity in practice and in theory. I ...
0
votes
1answer
28 views
Plane rotation homomorphism
I want to show that the following set (given group structure with matrix multiplication)
$$D=\left\{ D(\theta)=\begin{pmatrix}
e^{i\theta/2} & 0 \\
0 & e^{-i\theta/2}
\end{pmatrix}, 0 \leq ...
2
votes
1answer
20 views
Two dimensional euclidean motions
Let $\varepsilon_2$ be the set of Euclidean motions in $\mathbb{R}^2$, defined as the ordered pairs $(a,R(\phi))$ where $a \in \mathbb{R}^2$,
$$R(\phi)=\begin{pmatrix}
\cos(\phi)&-\sin(\phi) \\ ...
1
vote
2answers
47 views
Are the lengths from this recursive construction a geometric sequence?
In his 1999 review of Edward Tufte's Visual Explanations in the Notices of the AMS (third page), Bill Casselman gives a very pretty proof of the irrationality of the golden mean. More precisely, ...
1
vote
0answers
38 views
What is known about $n$ independent random variables that yield a “converse” to uniform sample of a coordinate from a surface of an $n$-sphere?
It's well-known that to sample a coordinate $(Y_1,\ldots,Y_n)$ from a surface of an $n$-dimensional unit-radius sphere, it suffices to generate $n$ independent random variables $X_1,\ldots,X_n$ from ...
1
vote
1answer
32 views
Projecting external points to a circle: Distance order preserving?
Given a circle, and a set of points $A$ that lie external to the circle; I perform the following simple operation:
I compute the point of intersection of the i) circle and the ii) line joining each ...
4
votes
1answer
98 views
Super hard Euclidean Geometry
The triangle $ABC$ is right angld at $A$. A line through the midpoint $D$ of $BC$ meets $AB$ at $X$ and $AC$ at $Y$. The point $P$ is taken on this line so that $PD$ and $XY$ have the same midpoint ...
2
votes
2answers
91 views
Euclidean Geometry: Diagonals of Cyclic Hexagon
Convex Hexagon $ABCDEF$ is cyclic. Prove that its three main diagonals $AD$, $BE$ and $CF$ are concurrent iff:
$$|AB| \cdot |CD| \cdot |EF|=|BC| \cdot |DE| \cdot |FA|$$
This seems like it might have ...
3
votes
2answers
87 views
Euclidean Geometry Area Problem
Let $\Gamma $ be the circumcircle of triangle $ABC$. Let $A_0$ be the center of the circle lying outside of $\triangle ABC$ and which is tangent to the segment $BC$ and to rays $\overrightarrow{AB}$ ...
1
vote
1answer
53 views
apollonian circles: why are radius and center dual?
This figure suggests the radii and centers (regarded as complex numbers) of the Soddy circles satisfy the same equation:
$$ a^2 + b^2 + c^2 + d^2 = \frac{1}{2} (a + b + c + d)^2$$
How can the circle ...
1
vote
1answer
39 views
Euclidean cirle question
Let $c_1$ be a circle with center $O$. Let angle $ABC$ be an inscribed angle of the circle $c_1$.
i) If $O$ and $B$ are on the same side of the line $AC$, what is the relationship between $\angle ...
2
votes
1answer
90 views
Derivative of circumradius
Let $P = \{p_1, p_2, \cdots, p_n\}\subset\mathbb{R}^2 - \mathbb{R}\times\{0\}$ be a set of $n$ distinct points. Define the function $f: \mathbb{R}\times\{0\}\to\mathbb{R}^+$ by
$$(t,0)\mapsto ...
7
votes
2answers
81 views
Find eigenspaces using ruler and compasses
I think this is an interesting question:
In the 2-dimensional real vector space, we are given a linear transformation $f$. Suppose we already know the images of the standard bases, say ...
3
votes
1answer
49 views
Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom?
While reading a paper (pdf) about the history of modern logic, I learned that some opinions (about deductive/axiomatic mathematics) typically attributed to David Hilbert can be traced back to Moritz ...
59
votes
4answers
1k views
Probability that a stick randomly broken in five places can form a tetrahedron
Randomly break a stick in five places.
Question: What is the probability that the resulting six pieces can form a tetrahedron?
Clearly satisfying the triangle inequality on each face is a necessary ...
0
votes
0answers
40 views
Given 2 outer points of a perfect circle, find the centerpoint
Alright, I hope this makes some sense.
I am using a software that can create arcs.
This arc is defined by:
Begin point
End point
Center of "circle"
The center is supposed to be the center of the ...
3
votes
2answers
107 views
Formal Proof that area of a rectangle is $ab$
I tried to prove that the area of a rectangle is $ab$ given side lengths $a$ and $b$.
The best I can do is the assume the area of a $1\times1$ square is $1$. Then not the number of $1\times1$ squares ...
0
votes
1answer
28 views
How to reflect a vector across a rotated plane?
Let's say I have vector $\vec{A}$ which has points (-5,6,3) and (4,-2,4). I also have plane $P$ which is defined by the three points, (4,-2,4), (5,-3,4), and (4,-3,4). My question is, how would I ...
1
vote
0answers
45 views
What's a non-standard model of Tarskian Euclidean geometry?
Tarski's axioms (see here: http://en.wikipedia.org/wiki/Tarski%27s_axioms) are a first-order axiomatization of Euclidean Geometry. Now, I believe the standard model for the axioms is the real number ...
0
votes
1answer
38 views
How to fit an object of constant size based on measurements to known points
I'm looking for a mathematical solution for solving where the base
of a camera crane (ie a constant square or rectangle of known dimensions) is
with measurements to known points. This seems to be a ...
2
votes
1answer
56 views
3D Geometry Proof by Contradiction /Contrapositive (high school)
Could someone evaluate my work?
A plane perpendicular to one of 2 parallel lines is perpendicular to the other line also.
My two column proof so far:
Let AB || CD and AB be perpendicular to plane ...
1
vote
2answers
104 views
Algebraic solution to find circle radius given distance of three external points from perimeter
I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is ...
5
votes
2answers
62 views
Inscribing a quadrilateral inside a rhombus
Let ABCD be a rhombus, its interior angles are $\alpha<\Pi/2$ and $(\Pi-\alpha)$.
Let w, x, y, z four points located respectively in (A,B), (B,C), (C,D), (D,A).
Suppose we have as inputs the ...
1
vote
2answers
31 views
Find point at which secant crosses circumference of circle
I don't know the proper terminology, or even how to draw the right diagram, but I'm looking to work out a for any given y.
...
1
vote
0answers
31 views
find example from some regularities. euclids elements
I heared that there are 6 basic math abilities.
and one of them is to find an example from regularities.
I want to develop this ability so I tried to find an example from real world.
but I am not ...
2
votes
1answer
110 views
Prove the similarity of isosceles triangles…
Two similar isosceles triangles are constructed outside of an parallelogram ABCD, the first being $ABB_1$ and second $CBC_1$ i.e. $|AB| = |AB_1|$ and $|CB| = |CC_1|$. Since $ABB_1$ and $CBC_1$ are ...
0
votes
3answers
52 views
Prove that $|PC|^2 + |PD|^2 = |AB|^2$ if
We have an angle of 90° so that there are 2 points A, B on each side of the angle, O is the vertex and |OA| = |OB|. On the arc AB with it's center being in O, we pick an arbitrary point P and draw a ...
0
votes
2answers
61 views
A question on Trigonometry (bisector)
If two bisector of a triangular is equal, then it is Isosceles triangular.
2
votes
3answers
108 views
Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space?
I have question.
Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space ? Please can you give me an advice some book names?
Thank you!
