Tagged Questions
0
votes
1answer
16 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 $\gamma$ ...
1
vote
1answer
48 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
40 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
61 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 ...
23
votes
1answer
493 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
87 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
26 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.
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 ...
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 ...
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
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
96 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
90 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 ...
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 ...
58
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 ...
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
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
96 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
61 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.
...
2
votes
1answer
108 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 ...
3
votes
3answers
72 views
inscribed angles on circle
That's basically the problem. I keep getting $\theta=90-\phi/2$. But I have a feeling its not right. What I did was draw line segments BD and AC. From there you get four triangles. I labeled the ...
0
votes
1answer
41 views
I do't understand how to do this problem and was wondering how to get the answer easily 180(n-2) n= the number of sides a figure has.
Please help with this poblem I nedd help with the basics of it.
$180(n-2) n$= the number of sides a figure has.
1
vote
1answer
67 views
We have a triangle $ABC$. On sides $AB$, $BC$, $CA$ choosen points $A_1$, $B_1$, $C_1$, different with vertices $A$, $B$, $C$.
We have a triangle $ABC$. On sides $AB$, $BC$, $CA$ choosen points $A_1$, $B_1$, $C_1$, different with vertices $A$, $B$, $C$.
Then for a specified $k_a$, $k_b$, $k_c$ we have
$\vec{BA_1} = ...
6
votes
2answers
100 views
Concurrency of A'L, B'M, C'N.
Need some help with the following problem.
Problem: In $\triangle ABC$ the midpoints of $BC$, $AC$, $AB$ are $L, M,$ and $N$ respectively, and the points on the circumcircle opposite to $A, B,$ and ...
0
votes
1answer
54 views
What is “degenerate” about degenerate quadratic surfaces?
In Wikipedia the table of quadratic surfaces is divided into 2 parts, the second being "degenerate quadrics". Why is this distinction made? and what does the word degenerate means in this case?
2
votes
2answers
81 views
Classification of Euclidean plane isometries
I suppose this question has already been asked here, but I cannot find it.
Is there any simple way to prove that there are 5 possibilities for isometries in the Euclidean plane? Namely: Identity, ...
7
votes
1answer
117 views
Rotation of $\mathbb{R}^3$ by using quaternion
Express the rotation of $\mathbb{R}^3$ by $\frac{\pi}{3}$ about the $x=y=z$ axis by using quaternions and identifying $\mathbb{R}^3$ with $(i,j,k)$-space.
Thoughts:
From my point of view, every ...
0
votes
0answers
40 views
Solving using Lambert Quadrilaterals
Let $l$ and $m$ intersect at $O$ at an acute angle. Let $A$, $B$ $≠O$ be points on $l$ and drop perpendiculars to $m$ from $A$ and $B$, intersecting $m$ at $A′$, $B′$. If $OA < OB$, kindly show ...
12
votes
2answers
183 views
6 point lying on a common circle
$Z$ is an interior point of segment $XY$. Three semicircles are drawn over segments $XY$, $XZ$ and $ZY$ on the same side. The midpoints of the arcs are $M1$, $M2$ and $M3$ respectively. A circle ...
1
vote
3answers
50 views
Are there any Heron-like formulas for convex polygons?
Are there any Heron-like formulas for convex polygons ? By Heron-like I mean formulas without angles as arguments and which takes as arguments only lenghts of sides of polygon - that is - we know no ...
4
votes
1answer
150 views
what are some isometries of S^2 without fixed points?
Spherical geometry question involving isometries.
Particularly looking for isometries with no fixed points.
3
votes
1answer
64 views
Why does aliasing cause loss of a degree of freedom in Euler angles?
I'm reading a book on 3D game math where the author points out that when using Euler angles the same orientation can be reached by doing two different operations; say rotating a cube 90 degrees around ...
1
vote
1answer
128 views
Existence of Gergonne point, without Ceva theorem
The intersection at one point (called Gergonne point) of the lines from vertices of a triangle to contact points of the inscribed circle can be proved immediately using Ceva's theorem.
Is there a ...
0
votes
1answer
107 views
proving parallel projection is onto
Background information- We are given two lines L and M and point p on L. We set up a correspondence from p<==>p' between the points of line L and M requiring segment PP'|| n for all p on line L. ...
3
votes
1answer
70 views
small circle inside embedding of complete graph in the plane
On the web, I found this beautiful drawing of the complete graph on 13 vertices:
It is on the Geometry Daily tumblr page. A computer scientist drew a more interactive version up to about 40 ...
0
votes
1answer
95 views
Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$
Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$.
I know the shortest distance exists between the curves on the common normal line. Is there any other shorter way to attempt?
5
votes
3answers
78 views
What characteristic of the triangle leads the the existence of the orthocenter
We all know that all three altitudes of a triangle meets in the orthocenter of the triangle. It's a quite classical problem and is proven.
However, what I really wanna know is what characteristic of ...
0
votes
1answer
30 views
A Statement About Points in the Real Euclidean Space
Suppose that $n \geq 3$, $x$, $y \in \mathbf{R}^n$, $d \colon= |x-y| > 0$, and $r>0$. Then how to prove the following assertions:
(a) If $2r>d$, there are infinitely many $z \in ...
-2
votes
1answer
70 views
A point inside a triangle and a property of the angles it makes.
Choose a point $P$ inside a triangle $ABC$. Draw $AP$, $BP$ and $CP$. Show that at least 2 of the 6 angles $PAB$,$PAC$,$PBA$,$PBC$,$PCA$,$PCA$ is less than or equal to $30^o$.
I know there's a simple ...
0
votes
1answer
68 views
Perimeter and area of a hexagon
The floor of a theater was built in the shape of a hexagon. The interior angles of the hexagon are the same and one side is 8 meters long. How long is the perimeter of the floor in feet?
What is the ...
2
votes
1answer
107 views
How to formally prove that this proof is (not) correct?
In lemma 2 in this article's section 5 there is a proof below and at the end it states that equation $\|(1-\alpha)p_j+\alpha p_i-p_k\|=\|p_i-p_k\|$ has only one solution $\alpha=1$.
Examples can be ...
4
votes
2answers
103 views
Congruent division of a shape in euclidean plane
Any triangle can be divided into 4 congruent shapes:
http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png
An equilateral triangle can be divided into 3 congruent shapes.
Questions:
...
1
vote
3answers
159 views
Euclidean Isometry
This is part of my homework problems, it is not for assignment nothing to hand in first of all, i just dont get how to go about this proof. If i can see it it would be good.
Question is
Show that a ...
0
votes
1answer
24 views
Conditions for the existence of a conical combination of some given vectors such that it lies in a cone?
Let $v_1,v_2,\dots,v_n,u_1,u_2,\dots,u_r\in\mathbb{R}^n$. Can one find analytical conditions (not write the problem up as a convex optomisation problem and argue it can be solved this way) under which ...
2
votes
1answer
61 views
How to choose $x$ evenly distributed points from within an n-ball
I would like to know how to choose $x$ evenly distributed points from within an n-ball. I think a formal way of defining this is that we want to choose $x$ points from within the n-ball such that we ...
