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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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
14 views

What do you call a projection of a hyperplane into a finite hypercube that keeps paraxial lines straight?

Similar to the Poincaré disc for hyperbolic space, is there a bijection from $\mathbb{R}^n$ into, say, $[-1,1]^n$, while any paraxial orthotope in $\mathbb{R}^n$ remains a paraxial orthotope after the ...
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1answer
36 views

Find Circle from Tangent Line and Two Points

I have two points a,b and a line L. I want an equation to find point c that is the center of the circle that touches a, b and L Thanks.
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54 views

How to prove this Gram determinant

Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$ \det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ...
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29 views

Find the angles defining an hyperspherical cap

For the hyperspherical cap of dimension $n+1$ find all the angle $\phi_1, \phi_2, \ldots, \phi_n$ which defines the cap? I mean, I know a cap is usually define by its height $h$ and its base $a$. ...
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2answers
84 views

Why are polyhedra related to the prime numbers 2, 3 and 5, but not to the prime number 7?

Just take a quick glance at all the numbers in these Wikipedia pages on polyhedra: http://en.wikipedia.org/wiki/Platonic_solid http://en.wikipedia.org/wiki/Archimedean_solid ...
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3answers
178 views

Smallest square containing a given triangle

Given a triangle $T$, how can I calculate the smallest square that contains $T$? Using GeoGebra, I implemented a heuristic that seems to work well in practice. The problem is, I have no proof that it ...
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54 views

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2. What is the smallest possible angle of the polygon? What is its smallest possible area? ...
2
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2answers
73 views

Area of Intersection of Circle and Square

Given a point $(x,y)\in [0,1]^2$ and $r > 0$, I would like to derive a general formula for the area of the intersection of the circle of radius $r$ centered at $(x,y)$ and the unit square. What is ...
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45 views

How to distribute a set number of points evenly in a rectangle?

So my motivation here is actually because I want to evenly distribute plants in a green house and ideally I would like to maximize the distance between the plants and the walls. It seems like there is ...
2
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1answer
61 views

How to prove $C=2πr$?

Everybody know this formula,but why the relation between $C$ and $r$ is linear relation? Not $C=2πr^{0.99}$ or $C=2πr^{1.01}$how to prove it,what axiom is it based on?
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17 views

Number of Lines formed by joining points formed by the intersection of lines in a plane

There are $\mathbf{n}$ lines in a plane no two of which are parallel. They intersect at $^{n}C_2$ distinct points in space. How many new lines are constructed by joining these points. The essential ...
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38 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
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27 views

Suppose that a parallelogram has vertices at $0, u, v$, and $u+v$. Show that its diagonals have directions $u+v$ and $u−v$.

I showed that the midpoint of $U$ and $V$ is $\frac12(U+V)$ then multiplied that by $2$ to get $U+V$, How would I get $U-V$?
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2answers
56 views

What is the geometric meaning of the transformation of R2/R3 when every vector is multiplied by −1? Is it a rotation?

I'd imagine a sphere with the center at the origin and all length of the vectors equals the radius. But I can't imagine what would happen if all the vectors is multiplied by -1, what would it be? I ...
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4answers
89 views

How to prove U•V = |U|•|V|cos(θ), if θ is the angle between |U| and |V|

This is a snippet from my book. How did they get from |U|$^2$ = U • V = |U|•|V| |U|/|V| ?
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1answer
31 views

We can tell that f is not a translation or glide reflection (hence, it must be a rotation). How?

Let A=(0,1) B=(0,0) C=(1,0) Suppose that f(A) = (0.4,1.8), f(B) = (1,1), and f(C) = (1.8,1.6). How do we prove that if its not a translate or glide, then its a rotation? Is it because since glide ...
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3answers
22 views

Reflect the plane in the $x$-axis, and then in the line $y = \frac12$. Show that the resulting isometry sends $(x,y)$ to $(x,y+1)$

I have a hard time proving this without using any numbers. How do I show that the point $(x,y)$ reflected across $y=\frac12$ is $(x, 1-y)$ ?
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0answers
11 views

Incentre and “circumcentre” without congruence criteria

friends! Paul Bernays says, in his supplements to Hilbert's Grundlagen der Geometrie, that the theorem that the bisectors of the angles and, respectively, the perpendicular bisectors of the sides, of ...
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1answer
135 views

Properties of sphere

Let $C$ be a circle with diameter $\overline{AB}$. Then it is well known that for any $P$ on the circle $C$ the angle $\angle APB =\frac \pi 2$. There are similar results for sphere?
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3answers
87 views

Bisector of angle formed at the orthocentre passes through the circumcentre

BdMO 2012 In an acute angled triangle $ABC$, $\angle A= 60$. We have to prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$ passes through the center of the ...
3
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1answer
85 views

Circumcircle of an isosceles triangle and length relation

I was asked to prove the following problem. Consider the following diagram where a triangle $ABC$ lies inside its circumcircle, $D$ is the point where the angle bisector $\alpha$ of $B$ intersects ...
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1answer
48 views

Show that a square with vertices t, u, v, w has center 1/4 (t+u+v+w).

I need a help with this question! Show that a square with vertices t, u, v, w has center 1/4 (t+u+v+w).
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2answers
43 views

Suppose that the tetrahedron has vertices t, u, v, and w. Show that the centroid of the face opposite to t is 1/3 (u+v+w)

I need some help with this question! Suppose that the tetrahedron has vertices t, u, v, and w. Show that the centroid of the face opposite to t is 1/3 (u+v+w), and write down the centroids of the ...
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0answers
28 views

Prove in a cyclic quadrilateral ${AC\over BD}={{ps+rq}\over pq+rs}$

Let $ABCD$ be a cyclic quadrilateral with length of sides $AB=p$, $BC=q$, $CD=r$, and $DA=s$. Show that $${AC\over BD}={ps+rq \over pq+rs}$$
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1answer
130 views

Proof of “Japanese Theorem” — Triangulation of Cyclic Polygon

On Mathoverflow, I saw this great result on the "Japanese Theorem". “Japanese Theorem” on cyclic polygons: Higher-dimensional generalizations? Given triangulation of a cyclic polygon, the sum of ...
3
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2answers
230 views

What is the history of the use of the term “scalene triangle”?

A "scalene triangle" is a triangle with three unequal sides. As far as I can tell, this term is not in much use in serious mathematics — in fact, before I became a high school math teacher, I'd ...
2
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0answers
26 views

Theorem about two quadrilaterals with parallel edges

I'm looking for a name for the following theorem: If $abAB$ lie on one line and $cdCD$ lie on another line, and furthermore $ac\Vert AC,ad\Vert AD,bc\Vert BC$, then $bd\Vert BD$. One can ...
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82 views

Heron's Formula; an Intuitive or Visual Proof

I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to show WHY the formula works. Do you know an intuitive ...
2
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38 views

How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
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1answer
31 views

is there a higher dimensional analogue of the first isogonic center?

I'm curious to know if, given four points $a, b, c, d$, you can always find a point $p$ such that last lines $pa, pb, pc, pd$ form equal angles pairwise. I'd also appreciate resources on 3d geometry ...
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1answer
90 views

abs(x)cos(x) in Fourier space

I am working on some problems concerning Fourier Transform and I am facing something I don't understand. I am trying to understand what is the representation of the function f(x)=abs(x)cos(x) in the ...
2
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2answers
28 views

Slicing up geometry to create triangles

When rendering object onto a screen, one must often cut up their objects into quads and triangles to allow the computer to process them and finally draw them onto a screen. I am trying to slice up a ...
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31 views

determine in what grid rhombus is a point

i have a rhombus ( i.e. diamond) grid determined by these equations ...
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30 views

A question of convergence

Let $T_1=[P_1,P_2,P_3]$ be a triangle of area one. Choose a point $P_4$ inside or on $T_1$ such that the area of triangle $T_2=[P_2,P_3,P_4]$ is half the area of $T_1$. Repeat this process in the ...
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35 views

Euclidean distance

I am studying a paper related to protein structure alignment where I find a equation to find the similarity of two nodes of a protein using a similarity measure $D$. This is given by $D(d1,d2) = 0.1 ...
2
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1answer
40 views

3 circles and 3 squares all inscirbed into a right angled triangle problem

This is quite a tricky question for me, but this is how far I got: My drawing may not be precise, but I do know the points of tangency. I am a little stuck now, and I would appreciate a great hint ...
4
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2answers
90 views

an olympiand geometrical problem

Let $ABCD$ be a convex quadrilateral and $E$ be a point on $CD$. Suppose $AC$ meets $BE$ at $F$, $BD$ meets $AE$ at $G$, and $AC$ meets $BD$ at $H$. If $FG//CD$, and the areas of $\triangle ...
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1answer
22 views

Find vector resultant in rhombus

Uhm I can't find a solution for this problem, perhaps someone can help me with a hint or a solution, thanks in advance :) $$DG=GH=HI=IG\\and\\ AE=EF=FB$$ Find resultant for U+V+W
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47 views

The Least Area For a Needle to Pass Through a Curve?

I don't know if this question is a famous one. One of my fellows asked me these questions to tease me, but I was able to find a solution for only one of these: There is one needle of length $2$ ...
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4answers
40 views

prependicular Vs prependicular bisector

We have $AH=HB$ and $BG=GC$ in the image below. Why is $AD=2\times FG$?
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53 views

Staircase Lemma

Let $S$ be a staircase-shape contained in the north-eastern quarter-plane. Let $k$ be the number of its south-western corners. In the staircase shown below, there are $k=4$ corners: In each corner ...
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Is it possible for Euclid's theorem to generate 3 4 5?

I am writing an algorithm to generate pythagorean triplets and I was going to use Euclid's theorem, however I have been unable to make it generate the first pythagorean triplet namely (3,4,5). Is this ...
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1answer
25 views

How to compute an angle in specific counter-clockwise direction between vectors

I have one incoming vector and multiple outgoing vectors in 2D. I need compute an angle in this way: Imagine an incoming vector parallel to the x-axis. Then the angle-value "starts" below the ...
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0answers
26 views

To construct a right triangle given the hypotenuse and sum of two legs [duplicate]

NOTE: I want a hint only. A compass and a straightedge construction:Given a hypotenuse and the sum of lengths of the legs,we need to construct a right triangle. MY TRY: From any ray $BE$, ,let ...
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1answer
53 views

Draw a picture of a cube with edges a+b, and show it cut by planes that divide each edge into a segment of length a and a segment of length b.

I am reading through 4 pillars of geometry and I need some help with this question. Draw a picture of a cube with edges a+b, and show it cut by planes (parallel to its faces) that divide each edge ...
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0answers
12 views

Updating vector components based on change of starting position

I have an xyz point, and a 3D vector originating at that point. I would like to be able to shift the starting xyz point and update the 3D vector accordingly. For example: Starting at the xyz point ...
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2answers
43 views

Maximize the inradius given the base and the area of the triangle

BdMO 2013 Secondary: A triangle has base of length 8 and area 12. What is the radius of the largest circle that can be inscribed in this triangle? Let $A,r,s$ denote the area,inradius and ...
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3answers
89 views

Equation of a line passing through a given point, perpendicular with a line [closed]

I need the equation of the line passing through a given point $A(2,3,1)$ perpendicular to the given line $$ \frac{x+1}{2}= \frac{y}{-1} = \frac{z-2}{3}. $$ I think there must bee some kind of rule ...
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3answers
97 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...