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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1answer
19 views

How to embed this circle tangent to the other circles?

I want to construct a circle that would be tangent to the $3$ circles and would have its diameter lie somewhere on the segment $BI$. What do you think is the best way of doing this?
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64 views

What's a bi-rhombus? [on hold]

Please, our class is desperate. We need to know what is a bi-rhombus. Our teacher asked us to do a graded project involving this
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2answers
20 views

What is the ratio of the side length of a regular hepatgon to the side length of the internal heptagon?

Given a regular heptagon with side length 1, create a star heptagon by connecting every vertice. Note that removing the "points" of the star yields a similar heptagon. I want to know the side ...
4
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1answer
38 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
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2answers
51 views

How can I find the volume of prism: $V = \frac{(a + b + c)Q}{3} $

In the book Handbook of Mathematics (I. N. Bronshtein, pg 194), we have without proof. If the bases of a triangular prism are not parallel (see figure) to each other we can calculate its volume by ...
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3answers
50 views

Show a complex equation has one or two roots

Let $a$ $\neq$ $0$, $b,$ and $c$ be complex constants. Show that the quadratic equation $az^2+bz+c=0$ has one or two roots. My thoughts: Let $a=a_1+ia_2,$ $b=b_1+ib_2,$ and $c=c_1+ic_2$. I also ...
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1answer
24 views

To use Vieta's formula for complex constant solution or not?

Let $b$ and $c$ be complex constants such that $z^2$ + $bz$ + $c$ = $0$ has two different real roots. Show $b$ and $c$ are real. I think I need to be using Vieta's formula, however I have solved it ...
0
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1answer
29 views

can you help me solve my menu board dilemma?

If I have a menu board that measures 35 3/8 $\times$ 71 5/8 and I need to cut in 3 equal pieces, what measurements should each piece be?
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1answer
41 views

The point of contact between a line with a circle

My question is: I have a circle of radius 40 and a line which the circle is tangent to. So, if I take a circle of radius 80, do the two circles have the same point of contact? I mean: do they (my ...
0
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1answer
47 views

How do 3 points define a plane?

I was solving a combinatorics problem which asked me to find the number of planes that can be constructed from a set of 25 points such that no 4 points in the set of 25 points are co-planar and then I ...
0
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2answers
36 views

Is it appropriate to apply Euclidean Distance to Complex Numbers?

Would complex numbers be considered as part of Euclidean Space? Would this measurement give an accurate result? If not, what would be a more appropriate distance measurement/similarity measure with ...
3
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1answer
39 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
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1answer
35 views

Roundness in Taxicab Geometry

I was just wondering whether circles are considered "round" still in taxicab geometry. I know that "roundness" is probably not a well-defined term and I know what a circle /appears/ to look like in ...
2
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3answers
48 views

Euclidean “straight line” calculation

Please see image first.. I have as input the following (I presume these are in effect Euclidean coordinates): The angle and the length of the red line. The angle and the length of the green ...
0
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1answer
26 views

Unique Euclidean isometry between affinely independent points

Let $u_0,\dots,u_n$ be vectors in $\mathbb{R}^n$ such that $u_1-u_0,\dots,u_n-u_0$ are linearly independent and similarly let $v_0,\dots,v_n$ be vectors in $\mathbb{R}^n$ such that ...
0
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2answers
31 views

Derive formula for coordinates of internal and external centers of similitude.

Given 2 circles $(x - x_1)^2 + (y - y_1)^2 = r^2$ and $(x - x_2)^2 + (y - y_2)^2 = r'^2$ (with radii $r, r'$) coordinates of the internal and external centers of similitude $C_i, C_e$ are given by ...
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4answers
68 views

Complex Numbers: Im$(\frac{12}{z-7})=1$

Sketch and describe the set of complex numbers satisfying $$Im(\frac{12}{z-7})=1$$ where $z=x+iy$ The answer should be in circle form. Here is what I have so far: $$Im(12)=z-7$$ $$Im(12)=x+iy-7$$ ...
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0answers
46 views

Banach Tarski paradox in Minkowski space

The Banach - Tarski paradox states: 'A three-dimensional Euclidean ball is equidecomposable with two copies of itself.' My simple question is: Does this paradox hold his validity also in the Minkowski ...
4
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1answer
393 views

IMO 2014 problem 3, first day

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such ...
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5answers
256 views

Tangent and angle bisectors

The tangent to the incircle of a triangle ABC is reflected about the external angle bisectors. Show that the triangle formed by the resulting 3 lines is congruent to ABC .
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2answers
461 views

How to make a perpendicular construction in 3 moves?

I've been playing Euclid: The Game for some time now. I'm quite addicited to it, trying to get all the records now. Suprisingly, I'm not able to get a record for some really early level. In Level 4 ...
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0answers
34 views

A mixtilinear tangency

Let ABC be a triangle with incircle $\gamma$ and circumcircle $\Gamma$. Let $\Omega$ be the circle tangent to rays $AB, AC,$ and to $\Gamma$ externally, and let $A^{\prime}$ be the tangency point of ...
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1answer
38 views

A construction of a triangle mapping with a homothety

Given an acute triangle $ABC$ draw a triangle $PQR$ such that $AB=2PQ,BC=2QR,CA=2RP$, and the lines $PQ,QR,RP$ pass through $A,B,C$ respectively. Note $A,B,C,P,Q,R$ are distinct. This is a problem ...
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26 views

Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
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1answer
24 views

euclid's 13th proposition

Euclid's 13th Proposition goes as : Proposition 13. If a straight-line stood on a(nother) straight-line makes angles, it will certainly either make two rightangles, or (angles whose sum is) equal ...
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1answer
48 views

Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
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2answers
31 views

Rotation of an equation

Problem: Suppose a line $L$ is given by the equation $\frac{x}{a} + \frac{y}{b}=1$, where $a$ and $b$ are non-zero real numbers. Let $\Re_{\frac{\pi}{2}}$ be the counterclockwise rotation of the ...
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1answer
26 views

Dilation proof verification

I am wanting to verify the proof below; can someone please tell me if they agree with the way I have argued this or if I have made any incorrect assumptions (and where they are). Thanks!! A ...
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1answer
44 views

Show every dilation is a non-constant linear function.

A dilation of reals is a function $f:\Re \mapsto \Re$ such that for some constant $c\neq0$ one has $|f(x)-f(y)|=c\ast|x-y|$ for all $x,y\in\Re$. Show that every non-constant linear function is a ...
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0answers
50 views

How to establish this result using induction?

A point $(x,y)$ in the plane is called a lattice point if both coordinates $x$ and $y$ are integers. Let $P$ be a polygon whose vertices are lattice points. Then the area of $P$ is $I + \frac{1}{2}B ...
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1answer
41 views

Square inscribed in a circle and an angle

Consider this badly drawn picture and the circle in the picture. Suppose that the circle has unit length, so that the area is $\pi$. Suppose that We know that the area enclosed by $ABCD$ is exactly ...
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1answer
28 views

Is taking the Euclidean norm of multiple Euclidean norms equivalent to taking the Frobenius norm?

I'm just a programmer venturing into the world of norms (is that even a thing?) here, and am wondering if two formulas are equivalent. Please forgive my ignorance! Suppose we have a $10\times3$ ...
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2answers
42 views

Express the length of the as a function of x

I am having problems understanding how to extract this information into a formula. ...
0
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1answer
15 views

What are the coordinates of a point on a rigid body after a rotation in 3D Euclidean space, given the initial coordinates and a center of rotation

Main question Let ($x_p$, $y_p$, $z_p$) be the initial coordinates of a point $P$ on a rigid body in a right-handed 3D Euclidean space. Let ($x_r$, $y_r$, $z_r$) be the coordinates of a center of ...
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80 views

A Modern Alternative to Euclidean Geometry

First of all, I want to master Geometry, I have knowledge on high school geometry and I was thinking of learning Euclidean Geometry. I bought a copy of Euclid's Elements, it is very interesting, ...
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40 views

Area of small trinagle in terms of Bigger triangle

I'm a school student and our teacher gave the question as follows: $A', B'$ and $C'$ trisects the sides of a triangle. then find the area of triangle $UVW$ in terms of triangle $ABC$. Please help ...
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2answers
65 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
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2answers
56 views

Why are these lines tangent?

I was trying the problems at http://euclidthegame.org and for level 20, ending up using, but couldn't see the reason behind the following: We have a circle centred on B and a point A outside the ...
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1answer
30 views

Proof about isometries, symmetry and reversing orientation.

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an isometry of the reals. Prove that $f$ is a symmetry around a point if and only if $f$ reverses orientation of $\mathbb{R}$. The orientation of ...
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2answers
2k views

Euclid the Game Level 2: Construct a line that bisects the given angle.

This is the level I'm trying to complete: My last idea was this, but I still don't get a message that I completed the level. Why is this not correct ? Last time I did mathematics was 10 years ...
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1answer
33 views

Proof about symmetry in isometries.

Suppose $f: \Bbb R \rightarrow \Bbb R$ is an isometry of the reals. Prove that $f$ is a symmetry about a point if and only if $f$ has a unique fixed point. Part 1: The assumption is $f$ is a ...
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3answers
35 views

Proof about isometries

Suppose $f\colon\mathbb R\to\mathbb R$ is an isometry of the reals. Prove $f$ is a non-trivial translation iff $f$ has no fixed points. Assumption: $f$ is a non-trivial translation (trivial ...
3
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0answers
50 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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0answers
16 views

Given a pair of conics, construct (synthetically) their shared tangent lines

There are various well-known ways to construct the common tangents to a pair of circles; this is an easy one. I also just learned that we can use Pascal's Theorem to construct a tangent to a conic at ...
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2answers
30 views

Is triangle congruence SAS an axiom?

I was wondering if there is a way to prove SAS in triangle congruence with Euclidean axioms. Thank you for your help!
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3answers
30 views

Euclidean geometry please help me

http://i62.tinypic.com/5osi8n.jpg please help me just wrote exam and wanted to know whether it was correct or not. I said (m+n)(m-n)=(m-n)^2 so I think it is not A right angled triangle what do you ...
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1answer
24 views

Find the equation of a plane that is perpendicular to another plane, parallel to a line and goes through a point

Find the equation of a plane which is perpendicular to the plane $$\pi_1\equiv x-3y-z+1=0,$$ parallel to a line $$l\equiv\frac{x - 2}{2} = \frac{y -3}{-3} = \frac{z}{1}$$ and goes through point $P ...
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3answers
61 views

Three Circles Meeting at One Point

We have three triples of points on the plane, that is, $X=\{x_1, x_2, x_3\}$, $Y=\{y_1, y_2, y_3 \}$, and $Z=\{z_1, z_2, z_3\}$, where $x_i, y_i, z_i$ are points on the plane. I was wondering if there ...
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0answers
19 views

Mapping between two unknown 3D coordinate systems from common motion

Coordinate systems A and B are rigidly linked in an unknown way. The platform then moves and the motion vectors [RA|TA] and [RB|TB] are calculated in each coordinate system. They are parallel but not ...
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2answers
45 views

Axis of rotation of composition of rotations (Artin's Algebra)

Say $R_1$, $R_2$ are rotations in $\mathbb{R}^3$ with axes and angles $(v_1,\theta_1), (v_2,\theta_2)$ respectively. Since $SO_3$ is a group, we have that $R_2 \circ R_1$ is a rotation with some axis ...