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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Equilateral and equiangular polygon

Can we have an equilateral polygon $n \geq 5$, which is not equiangular? Ot does every odd n-gon which is equilateral must be equiangular? Is a construction of an equilateral but not equiangular n-gon ...
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form of groups of motions of tessellations

I have read from the book "Mathmatics and Its History" by John Stillwell. In Section 18.6 it is about complex interpretations of geometry. The book says: The triangle and hexagon tessellations have ...
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Can the triangle inequlity extened to show the distance inequlity of a trapezium

$AB // CD$. What are the angle conditions (acute, obtuse or right angle) of $a,b,c,d$ to be satisfied the inequality $ |AB+BC| > |CD|$? $AB,BC,CD$ are distances.
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Do we need measure theory to prove length of diameter and circumference is constant is a circle?

In high school I was taught that $\pi$ is the ratio of the length of circumference and diagonal of a circle. But is it necessary to use some measure theory machinery to define the length of ...
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parallel and normal projections

I have a vector $v$ given by $(v_x, v_y, v_z)$ which makes an angle $\theta$ with the $x$-axis. The projection of $v$ onto $x$ is given by the dot product $$v\cdot x = ...
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When is a metric space Euclidean, without referring to $\mathbb R^n$?

Normally, the Euclidean space is introduced as $\mathbb R^n$. However, I've now been thinking about how one might define the $n$-dimensional Euklidean space only from the properties of the metric. ...
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How to prove $XP = X'P$?

Two triangles $\triangle ABC$, $\triangle A'BC$ have the same base and the same height. Through the point $P$ where their sides intersect we draw a straight line parallel to the base; this line meets ...
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justifying reflection across line in beltrami-klein model

Justify the following construction of the Klein reflection A' of A across m. Let Λ be an end of m and P be the pole of m. Join Λ to A and let this line cut y (which is the circle, my note) ...
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A curious near concurrence

In a triangle $ABC$ with incircle $\omega$, excenters $I_A, I_B, I_C$, let $t_{AB}$ and $t_{AC}$ be the tangent lines to $\omega$ through $I_A$ that are closer to $B$ and $C$ respectively. Construct ...
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153 views

Trigonometry and algebra question

Given: The total length of ad + dc The lengths of each ab, bc and ...
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Trigonometry and algebra question [duplicate]

Given: The total length of ad + dc The lengths of each ab, bc and ...
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proving that a point is a center of a circle

If i want to prove that a point $O$ is a center of a circle. is it sufficient to say that if $A,B,C$ are points On the circle and $AO=BO=CO$ so point $O$ is the center because of: Through any three ...
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Lamé parameters and distance on a curved surface

I was wondering if it is possible to compute the distance between two points which lay on a curved analytical surface. The surface is defined with differential geometry formulae (position vector of ...
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can I travel in the same direction along the surface of an ellipsoid without ever returning home?

Starting from an arbitrary point on an ellipsoid, moving straight at a random direction along the surface, are you always guaranteed to come back to the starting point eventually?
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Does this proof work to prove that the greatest area of a triangle inside a circle is when the triangle is equilateral?

Does this proof work to prove that the greatest area of a triangle inside a circle is when the triangle is equilateral? I gather it doesn't because most of the proofs I've seen use derivatives etc. If ...
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Equation of a line passing through a given point, perpendicular with a vector

Find the line that goes through A(1,0,2) and is perpendicular to r = (-2,3,4) + s (1,1,2) I did a bunch of work, but I don't know if any of it is right. I erased most of it, but this is what I came ...
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Finding distance from point to line

Knowing the position of 3 points($A, B, C$) , how can I get the distance from $A$ to the line $\overline {BC}$ if I know the angle?
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Is there a deep reason why $(3, 4, 5)$ is pythagorean? [closed]

The triple $(3, 4, 5)$ is a pythagorean triple - it satisfies $a^2 + b^2 = c^2$ and, equivalently, its components are the lengths of the sides of a right triangle in the Euclidean plane. But of ...
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Deriving the formula for the volume of a sphere

A circle $x^2 +y^2 =a^2$ is rotated about the $y$-axis to form a solid sphere of radius $a$. How do you express this motion mathematically in such a way that it allows me to arrive at the formula ...
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What is the ratio of the perimeter of $OPRQ$ to the perimeter of $OPSQ$?

Area of circle $O$ is $64\pi$. What is ratio of the perimeter of $OPRQ$ to that of $OPSQ$ ($\pi = 3$)? Okay i have tried couple of things but seems its not working . Please suggest me proper ...
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equation for a line in 3D, but with shifted origin

The (unit) direction vector $r$ of a line $L$ in 3D I write in spherical coordinates, $$ r_x = \sin\theta\cos\phi \\ r_y = \sin\theta\sin\phi \\ r_z = \cos\theta $$ This line passes through the point ...
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generalizing symmetry axis of elliptical contours in 3D

The contours of the following function $f$ trace out an ellipse, $f(x, y, z) = \exp(-x^2a)\exp(-y^2b)$, where $a\neq b$ are positive, real constants greater than zero. The axis of these ellipses is ...
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Probability of a certain circular configuration

Pick each of $n$ angles , $\theta_1$ through $\theta_n$ , uniformly randomly in the range $[0,2\pi$]. Define the distance $d_{i,j}$ between $\theta_i$ and $\theta_j$ by $d_{i,j} = \min(|\theta_j - ...
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Why is it impossible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that $|z_1- z_2|=|z_1-z_3|=|z_2-z_3|=|z_1-z_4|=|z_2-z_4|=|z_3-z_4|$?

A. It is possible to find distinct $z_1,z_2,z_3\in \mathbb C$ such that $|z_1-z_2|=|z_1-z_3|=|z_2-z_3|$. Answer: True B. It is possible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that ...
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Sum of distances from triangle vertices to interior point is less than perimeter?

Let $M$ be a point in the interior of triangle $ABC$ in the plane. Prove $AM+BM+CM<AB+BC+CA$. The above question was posed to someone I know who is taking high-school Euclidean geometry. I'm ...
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466 views

Do the tangents of two circles define concentric circles?

Given two non-overlapping circles, $R_1$ and $R_2$. The radii of $R_1$ and $R_2$ may be different. The distance between the centers of $R_1$ and $R_2$ is defined as $x$. Draw the four tangents ...
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612 views

Finding the x-coordinate of the max point of $y = x\sqrt {\sin x} $ so that it satisfies the equation $2\tan x + x = 0$

The maximum point on the curve with equation $y = x\sqrt {\sin x} $, $0 < x < \pi $, is the point A, Show that the x-coordinate of point A satisfies the equation $2\tan x + x = 0$ I ...
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Euclidean Geometry

How i can give a fast proof of the following fact: Given four points on $\mathbb{R}^3$not contained in a plane we can choose one such that its projection to the plane passing trhough the others is in ...
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Circle Chord Sequence

This is my first post, so be nice! When I was in my first Geometry class in high school, I asked the teacher the following: Given a circle of radius 2a, find the length of the chord running parallel ...
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Can a tetrahedron lying completely inside another tetrahedron have a larger sum of edge lengths?

Find 2 tetrahedrons $ABCD$ and $EFGH$ such that $EFGH$ lies completely inside $ABCD$. The sum of edge lengths of $EFGH$ is strictly greater than the sum of edge lengths of $ABCD$. I am completely ...
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3D Geometry Question

In $3$-dimensional Geometry, if angle made of line segment $OP$ with $X,Y,Z$-axis are in $1:2:3$, then what is the angle made by line segment with $Y$-axis? My Solution: Let $\alpha,\beta$ and ...
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Let $Ax = b$ be a system of hyper-planes that form a bounded convex $D$. Can $D$ be partitioned into union of adjacent simplices?

Let $Ax = b$ be linear system that forms $q$ bounded region $D$. If the columns of $A$ are independent, can $D$ be written as a union of adjacent simplices?
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Curious basic Euclidean geometry relation

I was plotting some equations and I got with the curious relation If we build the triange Such that it follows the following relation: $$AD=a$$ $$DB=b$$ $$AC=ak$$ $$CB=bk$$ Then when we vary $k$ ...
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Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...
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Probability that points are on a straight line

I am looking at a formula to calculate the probability that $n$ points are on a straight line between point $1$ and point $n$ in 2d Euclidean space. If the points are exactly on the line, the ...
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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 $\gamma$ ...
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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, $$ ...
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Using Square area finding quadrilateral area

Area of square ABCD is 169 and that of square EFGD is 49. Find area of quadrilateral FBCG I am stuck and just thinking which way can be helpful for me finding this area of quadrilateral FBCG. ...
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what is the area of face of the cube in $m^2$

A fly is trapped inside a hollow cube. It moves from A to C along edges of cube, taking shortest possible route. It then comes back to A again along edges, taking longest route(without going over any ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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About Euclid's Elements and modern video games

Update (6/19/2014) $\;$ Just wanted to say that this idea that I posted more than a year ago, has now become reality at: http://euclidthegame.com/ 12.292 users have played it in 96 different ...