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.

learn more… | top users | synonyms (1)

0
votes
1answer
29 views

arithmetic average over the spherical surface?

intuition behind taking arithmetic average over the spherical surface? . wiki definition :- Consider an open set $U$ in the Euclidean space $R^n$ and a continuous function $u$ defined on $U$ with ...
1
vote
1answer
20 views

the inner product of 2 vectors in complex vector space

Let us consider two vectors $u$ and $v$ are in the complex vector space. The inner product of these two vectors is defined by $u\cdot v=(u_1^*\cdot v_1, u_2^*\cdot v_2,\cdots\cdots, u_n^*\cdot v_n)$ ...
4
votes
0answers
49 views

Archimedes Classic Proof for Area of Circle: Love it but can't grasp one aspect…

The proof assumes that:... The perimeter of any CIRCUMSCRIBED regular polygon is GREATER than the circumference of the circle. ie: !http://www.themathpage.com/atrig/Trig_IMG/eval1.gif Is this an ...
1
vote
0answers
57 views

Max value of $9\lambda^2 -2 \mu^2$

Suppose vector $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ such that: \begin{align} \lvert \mathbf{a} \rvert &= \mu \lvert \mathbf{b} \rvert \\ \lvert \mathbf{c} \rvert &= \lambda \lvert ...
0
votes
0answers
36 views

Finding the area of the different portions of a rectangle that lie in a grid.

I am an undergraduate student working on a large research project and one part involves calculating the portions of a rectangle that lie in different parts of a cartesian grid. In the figure below, I ...
0
votes
3answers
28 views

Number of chords of a circle having natural length

In a circle of radius 17, point p lies on a distance 15 from center.How many distinct chords of this circle passing through p do have a natural length? I tried to use the notion of Power of point ...
2
votes
0answers
18 views

Do non-trivial closed and bounded convex sets with this property exist?

Suppose that $C$ is a closed and bounded convex set which is a subset of euclidean plane and which has the property that for every three points $P,Q,R$ which are on the boundary of $C$ the circle ...
0
votes
0answers
10 views

maximal volume/diameter of a set of simplexes

I am trying to develop a simplicial integral in $R^n$ and I am faced with the problem of controlling the "compacity" of a set of simplexes: Let $S$ be a finite set of n-d simplexes in $R^n$. Define ...
1
vote
2answers
54 views

Can $n$ circles be drawn such that all have a common intersection but no two intersect individually

I was fiddling with plane geometry when a question came into my mind: Can $n$ circles ($n \ge 3$, $n \in \mathbb{N}$) be drawn such that: $C_1 \cap C_2 \cap C_3 \cap \ldots \cap C_n \not = ...
0
votes
0answers
48 views

USAMO 2005, Problem3 (Triangle Geometry)- Is my solution correct?

USAMO 2005, Problem 3: Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is ...
0
votes
0answers
32 views

Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
1
vote
2answers
40 views

New Golden Ratio Construct: which one of my constructs is superior/simplest--squares & circles or just circles?

I have found yet another golden ratio construction. Geogebra gives it the value of 1.61803398874990 to the ratio between the yellow and blue lines in the figure below, which is the golden ratio PHI. ...
3
votes
1answer
42 views

Found a New Golden Ratio Construction with Equilateral Triangle, Square, and Circle. Geometric/Trigonmetric proof?

The below figure discloses a new golden ratio construction with an equilateral triangle, square, and circle. Geogebra gave me the value of the golden number 1.618 for the ratio of the yellow line to ...
2
votes
1answer
40 views

maximal volume/diameter of polyhedron

I am trying to develop an integral in $R^n$ and I am faced with the following problem: Given a polyhedron $P$ in $R^n$ of diameter d, define the "compactness" of $P$ as the quotient of the volume of ...
1
vote
0answers
37 views

An interesting geometry problem with angle bisectors and tangent

I have found the following problem: There is an acute $\triangle ABC$. Denote its circumcircle as $\omega$. The angle bisector of $\angle BAC$ intersects $BC$ and $\omega$ in points respectively $A_1$ ...
1
vote
2answers
50 views

The Golden Ratio in a Circle and Equilateral Triangle. Geomertic/Trigonometric Proof?

Geogebra gave me 1.61 for the following Golden Ratio construction shown below. Firstoff, has anyone seen anything similar to this construction? Basically begin with an equilateral triangle. ...
2
votes
1answer
35 views

Distance between incentre and orthocentre.

I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius. I considered $\triangle API$ ($P$ is ...
0
votes
0answers
16 views

How does one get $p=2$ from a condition that there be non-trivial linear transformations of every dimension that to any power are $p$-norm-preserving?

Verifying that (p=2) satisfies $$\forall n\in\mathbb{Z}^+.\exists A\in(\mathbb{R}^{n\times n}\setminus\{I_n\}).\forall k\in\mathbb{R}.\forall ...
-3
votes
0answers
48 views

The Golden Ratio in a Circle, Triangle, and Square: simple geometry/trigonometry construction.

I believe that I have found the golden ratio in the below figure. Geogebra is saying it is very close. Might anyone have a geometric/trigonometric proof? An equilateral triangle ABC is inscribed in a ...
1
vote
2answers
40 views

Confusion on wording of an elementary geometry problem

I really want to know the following geometry problem is valid or not. (Please don't change the wording of the problem. Please answer it is valid or not. Please answer frankly.) "ABCD is a ...
1
vote
1answer
39 views

Prove any line passes through at least two points

I've started reading Introduction to Algebra by Cameron, and I'm stuck on the first exercise. Q. Prove any line passes through at least two points using the axioms given below. Definitions: ...
0
votes
0answers
31 views

How to find all those points whose distance from $x=(2,0)$ is minimum, using $\|x\|=|x_1| + |x_2|$?

The points must be in the closed ball $\{y : \|y\| \le 2\|x\|\}$. I know $|y_1|+|y_2|$ needs to be $\le 4.$ Other than that, I am confused about how to find all the points that are minimum distance ...
1
vote
0answers
25 views

Rewrite each isometry as the composition of at most three reflections

Write each of the following isometries as a composition of at most three reflections: $\rho_{(1,0),\frac{\pi}{6}}$ $\tau_{(1,0)+(0,1)}=\tau_{(1,1)}$ $\sigma_{l_{BC}} \rho_{(1,0),\frac{\pi}{6}} ...
0
votes
0answers
12 views

Prove that $l$ and $l_{AB}$ are parallel if and only if $\sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id$

Prove that $l$ and $l_{AB}$ are parallel if and only if $\sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id$ I imagine that this proof has to be along the lines of a proof by contradiction, ...
0
votes
3answers
60 views

Assuming that the sum of the angles of any triangle is 180, how can I deduce Euclid's 5th postulate?

I already did the reverse, namely, if we assume Euclid's 5th postulate, then the sum of the angles of any triangle is 180 degrees. Now I need to show the converse, but I don't really know how to ...
1
vote
1answer
62 views

Geometric inequality $180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}$

Point $M$ is located inside a regular $n$-gon. Prove that there exist different vertices $A$ and $B$ that $$180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}\le 180^{\circ}$$ My work so far: ...
1
vote
1answer
51 views

Simple Golden Ratio Construction with Three Lines, and Interesting Implied Circle?

Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular ...
3
votes
4answers
60 views

Golden Ratio Conjecture in three simple Geogebra shapes--circle, triangle, and square.

A circle, equilateral triangle, and square of equal heights are all placed on the same horizontal line as shown below. The circle is tangent to the triangle which is centered upon the left edge of ...
-1
votes
0answers
34 views

PHI Conjecture: Fibonacci Number Line Segments & Simple Golden Ratio Puzzle?

This is a golden ratio conjecture. Does the golden ratio exist as conjectured below? The three line segments in the figure below have lengths of the Fibonacci numbers 2, 3, and 5. blue=2 green=3 ...
3
votes
1answer
70 views

Two Equilateral Triangles and the Golden Ratio: Simple Geomtric Proof

Two equilateral triangles rest upon the same horizontal line, as shown in the linked figure below. The red triangle is tilted so that one corner touches a side of the black triangle at the midpoint ...
0
votes
1answer
20 views

Is this definition of a Euclidean frame well-defined?

Going through my lecture notes on geometry I find a definition of a Euclidean frame which doesn't seem to have been formed correctly (most likely written down wrong). So I've taken it upon myself to ...
2
votes
1answer
29 views

Angles in Hilbert's axioms for geometry

In Hilbert's axioms for geometry, the following elements are presented as undefined (meaning "to be defined in a specific model"): point, line, incidence, betweenness, congruence. In deed, when ...
0
votes
1answer
17 views

How to fit a convex quadrilateral inside another short of cutting them out and playing with them?

I have two convex quadrilaterals (ABCD and WXYZ). Their sides and their interior angles are known. I also know that WXYZ fits perfectly inside ABCD with each corner point touching a different side. ...
2
votes
0answers
113 views

Water filling problem in Blocks - Algebra Question

Consider a rectangular plot comprising $n\times m$ square cells on which $nm$ cement blocks of various heights are stored, one block per cell. The base of each block covers one cell completely, and ...
1
vote
2answers
47 views

Power of a point proof

I found the question on page 13 of this link. Let $P$ be a point inside a circle such that there exist three chords through $P$ of equal length. Prove that $P$ is the center of the circle. I ...
4
votes
3answers
147 views

Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
-1
votes
1answer
22 views

Isometries in $\mathbb{E}^2$

We define $\mathbb{E}^2 = (\mathbb{R}^2, d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{R}^2$. We say that a function $f:X \rightarrow Y$, where $X, Y$ are metric spaces, is an isometry ...
1
vote
1answer
29 views

An elementary problem in Euclidean geometry

Let $ABC$ be an acute triangle ($AB < AC$) which is circumscribed by a circle with center $O$. $BE$ and $CF$ are two altitudes and $H$ is the orthocenter of the triangle. Let $M$ be the ...
1
vote
1answer
62 views

Draw A Triangle From 3 Excenters and Ex-radii

My teacher gave me this problem and told me to think- " Is it possible to draw a triangle, given the three ex-centers and length of the ex-radii?" I don't know if it's possible or not. So, my ...
0
votes
1answer
58 views

How to show $\partial A = \varnothing \Rightarrow A=R^n$ [closed]

Let $A\subset R^n$ and dim$A=n$, $\partial A$ is the relative boundary of $A$. If $\partial A=\varnothing$ how to show $A$ is $R^n$ ? Picture below is from XX page of Schneider R.-Convex Bodies_ ...
-1
votes
2answers
31 views

Prove $∠ADM = ∠ACB$ of triangle $ABC$ [closed]

Suppose that $ABC$ is a triangle. Let $D$ be its circumcenter and let $M$ be the midpoint of $\vec {AB}$. Show that $∠ADM = ∠ACB$.
1
vote
1answer
25 views

Every orthogonal matrix represents a rotation around an axis

Is it true that every element of the group $O(n)$ represents a rotation around some axis? I'd like this to be true in order to decompose any matrix $R \in O(n)$ as a block matrix in $O(n-1)$ and a 1 ...
0
votes
0answers
12 views

How to determine the angle from a point and the plane tangent points in a sphere

I have an UAV modeled in three dimensions with let's say position coordinates $p_{uav} = (x_1,y_1,z_1)$ that is moving in a direction $d = (d_x,d_y,d_z)$ and a moving obstacle modeled as a sphere with ...
3
votes
1answer
65 views

How to prove a regular pentagon is formed by knotting a rectangular strip of paper?

I found this interesting problem from a friend (From Arthur Engel's-Problem Solving Strategies book). The method to begin the problem is as follows- Step 1.Take a rectangular strip of paper ...
4
votes
1answer
23 views

There are finitely many $n$-dimensional wallpaper groups

There are $7$ Frieze groups, $17$ wallpaper groups and $230$ space groups, which are the 1,2,3-dimensional cases of isometries on $\Bbb R^n$ (I think? Frieze groups seem to require $[0,1]\times\Bbb R$ ...
1
vote
0answers
27 views

Prove a harmonic range from a familiar picture

During solving some simple problem (10th grade), I found this interesting problem, which I got no clue to solve it clean and properly. Hope someone can give me some hint to solve it. Thanks. Given ...
0
votes
0answers
18 views

Are given maps isometric?

I'm trying to determine if certain maps are isometric in $\mathbb R^2$. The two maps I have to analyze are f such that: $|f(X)| = |X|$ $f(X)*f(Y) = X * Y$ where $*$ is the dot product (inner ...
0
votes
0answers
27 views

Convert a 3d vector into a rotation matrix?

Is it possible to compute a Rotation matrix given a 3d vector given in the Euclidean space? and if not what would it need? An illustration of my situation. Illustration of my problem I have a ...
25
votes
5answers
2k views

Right triangle inscribed in a square. Find the square area?

I hope it's valid to ask for (a more neat solution) of a problem on this network, despite the fact that I don't have a strict definition of the word "neat". Here is the square and the right triangle ...
0
votes
0answers
15 views

How do i convert an x,y,z to an Q configuration?

I am trying to implement a tracking application for a robot arm, which purpose is relocate itself based on the position of an object seen from the tool point. illustration: http://imgur.com/5oojXdh ...