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)

-2
votes
0answers
16 views

To find/create midpoint, is it easier to bisect a line segment, or double a line segment? With only compass and ruler / straightedge. [on hold]

Suppose one wants to find the midpoint of a line segment. Is it generally easier to simply draw two lines of equal length end to end, or is it easier (does it count as less steps) to draw a line and ...
1
vote
1answer
21 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 ...
2
votes
4answers
36 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
23 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
54 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 ...
1
vote
1answer
17 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
22 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
16 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
59 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
37 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
101 views

What figure inside a circle is seen at a right angle from every 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
vote
0answers
31 views

Finding a relation between three points in a small circle of a sphere

I have a relation as follows. I am given two points $C, D$ on a circle, and a point $P$ somewhere inside of it. I would like to find the quantity $|AC| |AD|$, where $A$ is a point on the circle lying ...
-3
votes
3answers
55 views

A simple geometrical question regarding three circles and a line. Trigonometric construction. [on hold]

In Figure 1 three tangential circles all have the radius of 1 or r. What is the ratio of the blue line to the yellow line in terms of r, and in terms of r=1?
0
votes
1answer
14 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
24 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
60 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
57 views

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

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_ ...
0
votes
2answers
25 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
9 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
59 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
votes
0answers
27 views

A geometric inequality about the internal besectors [on hold]

prove that in every triangle the following inequality is hold: $$\frac{1}{w_\alpha}+\frac{1}{w_\beta}+\frac{1}{w_\gamma}\le \frac{2}{\sqrt{3}}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ ...
1
vote
0answers
26 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
17 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
26 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 ...
-2
votes
0answers
15 views

Area of a triangle determined by an interior point of a circle [closed]

Any hint is welcomed: Let $ABC$ an equilateral triangle inscribed in the unit circle $\mathcal{C}$. Let $P$ an interior point of the circle $\mathcal{C}$ and denote by $\Delta_{P}$ the triangle whose ...
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 ...
2
votes
1answer
41 views

Prove that $DQ \times DB = DP \times DC + DR \times DA$.

Let $ABCD$ be a parallelogram, with $P$, $Q$, and $R$ the points on which a given circle passes through $D$ and cuts through the segments $CD$, $BD$ and $AD$ respectively: How do you prove that $DQ ...
0
votes
1answer
47 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a ...
0
votes
1answer
30 views

Moving Line Segment Problem part 2

This question is related to a question I asked a while ago here on math.stackexchange: Moving Line Segment Problem The rules for how the line segment can be moved are the same: The endpoints must ...
0
votes
0answers
11 views

Isogonal Conjugate of point outside of triangle

I was wondering about reflections of lines over the external bisectors instead of external bisectors in a triangle. Here is a problem that brought it up: Let $P$ be a given point inside quadrilateral ...
3
votes
0answers
25 views

Volume of a cone [duplicate]

I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: ...
0
votes
1answer
28 views

Prove perpendicular bisectors of non-parallel lines intersect

Suppose that $A$, $B$ and $C$ are points and that $AB$ and $BC$ are not parallel. Show that the perpendicular bisector of $AB$, $l$, and the perpendicular bisector of $BC$, $l'$, are not parallel and ...
0
votes
0answers
20 views

Prove Concurrency using Radical Axis of Circumcircles

Let the incircle of $\triangle ABC$ touch sides $BC,CA,AB$ at $D,E,F$, respectively. Let $\omega,\omega_1,\omega_2,\omega_3$ be the circumcircles of $\triangle ABCm,\triangle AEF,\triangle ...
6
votes
2answers
169 views

Optimal path around an invisible wall [duplicate]

The Problem On an infinite plane there are two points, $A$ and $B$, a unit distance apart. There is a $50\%$ probability that there is an invisible wall somewhere between the two points. The ...
0
votes
1answer
13 views

Characterizing isotropic measures

A Borel measure $\mu$ on $S^{n-1}$ is called isotropic if $$\int_{S^{n-1}} \langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$ for all $\theta\in S^{n-1}$. This means that in ...
2
votes
3answers
56 views

Axioms of Geometry?

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ...
0
votes
2answers
24 views

Prove that DE || BC

Let M be the midpoint of side BC in triangle ABC. The angle bisector of BMA intersects AB in D, while the angle bisector of CMA intersects AC in E. How can i prove that DE||BC? I drew out the ...
1
vote
1answer
31 views

Finding the set of points on the sphere with an equal product of distances

Given two points $x_1$ and $x_2$ on the sphere, one can find another set of points on the sphere $\{y_1, y_2\}$ such that the product of Euclidean distances to the given points $x_i$ is the same for ...
0
votes
0answers
23 views

Finding the number of lattice points enclosed by a right triangle with non-integer coordinates.

I would like to find the number of lattice points enclosed by a triangle with coordinates $(0,0)$, $(0,a)$, and $(b,0)$. I Assumed that I would merely need to truncate $a$ and $b$, and apply and use ...
0
votes
0answers
23 views

How many Pascal hexagons can I construct with 6 different points on a circle?

I have a basic knowledge about combinatorics and I am in a euclidean geometry class. My question is : How many Pascal hexagons can I construct with 6 different points on a circumference? It could ...
1
vote
1answer
35 views

Name of the geometric figure of points ${\bf x} \in \Bbb R^n$ with $1$-norm $||{\bf x}||_1 = 1$

Is there a name for the figure $$\{{\bf x} \in \Bbb R^n : ||{\bf x}||_1 = 1\} \subset \Bbb R^n ?$$ Things like this seem to usually have names, for instance, the $n$-cube or $n$-ball. In $2$ ...
2
votes
5answers
68 views

what is the value of angle A

The triangle ABC is random. The line $AD$ is twice big as the line $DC$ ($AD=2*DC$). We know only the two angles that are shown in the picture. What's the value of angle $A$?
1
vote
1answer
28 views

Median BM of triangle ABC two results

Given Calculate the measure of the median $\overline{BM}$ of ABC triangle, given A (-6.1); B (-5,7) and C (2,5) I get this result: $Xm = \frac{Xc - Xa}{2} + Xa$ $Xm = \frac{2-(-6)}{2} + (-6) = ...
19
votes
11answers
5k views

In a right triangle, can $a+b=c?$

I understand that due to the Pythagorean Theorem, $a^2+b^2=c^2$, given that $a$ and $b$ are legs of a right triangle and $c$ is the hypotenuse of the same right triangle. However, most of the time, ...
0
votes
1answer
36 views

Is $f(x) = x^2$ a scalar function?

Take a simple parabola. It is a function that has a one-dimensional co-domain $$f(x) = x^2 $$ It is mapping the set of values in its domain, to one-dimensional values in its co-domain, and it ...
4
votes
1answer
65 views

Why this function describes a euclidean ball?

In Stephen Boyd's convex optimization book at page 97, one can read : $$ a,b \in R^n $$ $$ (1-\alpha^2)x^Tx-2(a-\alpha^2b)^Tx+a^Ta-\alpha^2b^Tb \leq 0 $$ is convex (in fact a euclidian ball) if $ ...
0
votes
1answer
17 views

Find distance of overlapping squares

How to find distance center to center from square $1$ to square $3$, if we need overlap area is $15.46 mm^2$. if we know each side of the square is $6.9 mm$. Firstly I find the distance is $9.3 mm$ ...