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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Tournament of Towns Geometry Problem, Proof by Construction?

I was to prove the following proposition from an old Tournament of Towns problems archive: Problem. A circle $\omega_{1}$ with center $O_{1}$ passes through the center $O_{2}$ of another circle $\...
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61 views

A simple proof that a polygon circumscribing a circle overestimates its perimeter

Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side ...
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71 views

Rectangle inscribed in a circular sector of angle 60

My apologies if this has been asked before. Given a circular sector, say of radius $r$, with internal angle $60^{\circ}$, construct a rectangle inscribed in that sector so that the length of the ...
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27 views

Euclidean Geometry Equilateral Triangle Problem

ABC is a equilateral triangle with vertex A fixed and B moving in a given straight line. Find the locus of C. Though it is clear that being an equilateral triangle, the size of the triangle must ...
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Is it true that: $\| a \| \| b \| \cos \alpha = \langle a,b\rangle$ [closed]

Let $a , b \in \mathbb{R}^n$ and let $\alpha$ be the angle formed between $a$ and $b$. Is it true that: $$ \| a \| \| b \| \cos \alpha = \langle a,b\rangle $$ ($\langle\cdot,\cdot\rangle$ being the ...
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Distance between a plane and a point

I understood that for finding a distance between a plane and a point we first find a vector between a point on a plane and the given point and then take the projection on the normal vector. Is $D=\...
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2answers
30 views

How do i compute the closest points on a sphere given a point outside the sphere?

I looking for method which can compute the yellow area in this image.. The ball with the green fill is a sphere, where i know the center point and the radius of it. The circle with the red fill ...
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1answer
78 views

What is the best way to inscribe a golden rectangle into a pentagon? Do more golden ratios emerge?

Below I drew a golden rectangle in a pentagon in Adobe Illustrator. What would be the best way to inscribe a golden rectangle into a pentagon as shown in the figure below in a mathematical manner? ...
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1answer
19 views

Distance between two lines

Find the distance between the lines $l_1:$ $x=1+4t,y=5-4t,z=-1+5t$ and $l_2:x=2+8t,y=4-3t,z=5+t$ So the approach in general is to find a vector that is orthogonal to 2 planes that the lines are in ...
2
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1answer
32 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 ...
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16 views

Checking if vector crosses the simplex

Let assume that I have a point in $x \in \mathbb{R}^n$ Also I have a non-zero vector defined by it's endpoint attached to this point. The third thing I have is a simplex of $\dim=n$, such that the ...
2
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1answer
36 views

Show that if a convex polygon is entirely inside another convex polygon, then the outer polygon has perimeter greater than or equal to the inner one.

Can anyone help me out with proving this statment? "Show that if a convex polygon is entirely inside another convex polygon, then the outer polygon has perimeter greater than or equal to the inner ...
2
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1answer
38 views

Proof/justification that a circumscribed regular polygon has a perimeter greater than the circumference of the circle?

According to Archimedes, 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 This does seem ...
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1answer
43 views

How can I find ratio between area of triangle and area of quadrilateral?

I have a parallelogram $ABCD$. $E$ is center of $AD$. $O$ is center of $AC$ and center of $DB$. $F$ is the intersection point between $CE$ and $DO$. Point $G$ is the intersection between $EO$ and $AF$....
2
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1answer
41 views

Minimum Perimeter of a triangle

I have been playing the app Euclidea, I have been doing quite well but this one has me stumped. "Construct a triangle whose perimeter is the minimum possible whose vertices lie on two side of the ...
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51 views

Defining perpendicular lines in the 3D space

Is there a universal agreement about the definition of "perpendicularity" between two stright lines in the 3 dimensional euclidean space? Do they need to meet or it is enough to have perpendicular ...
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3answers
63 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 ...
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2answers
33 views

Prove that if $A \perp B$ and $B \perp C$, then $A \parallel C$

Suppose that $A$, $B$ and $C$ are non-zero vectors in $\mathbb R^2$. Show that if $A$ and $B$ are orthogonal and $B$ and $C$ are orthogonal then $A$ and $C$ are parallel. I feel like this should be ...
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1answer
585 views

Online tool for making Geometric Constructions.

There was a website where it tasked you making different geometric shapes using only a compass and straightedge. I've looked for it and I can't find it or even discussion about it. What I do remember ...
2
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1answer
19 views

General conic equation and coefficient matrices

For a general conic $Q(x,y)=ax^2+2hxy+by^2+2gx+2fy+c$ we define a matrix $A$ as follows: $A=\left( \begin{matrix} a& h& g\\ h& b& f\\ g& f& c\end{matrix} \right)$. Then we ...
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Covering annulus with $2d$ symmetric pieces

Let $g:\mathbb R^d \to \mathbb R$ be a nonnegative radial function and $s\ge1$, $n\in\mathbb N$ be given. I want to check that the following inequality holds: $$ \frac{1}{2d} \int_{s<|y|<s+n} g(...
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1answer
15 views

Space geometry problem including centroid of a pyramid

Let $VABC$ be a triangular pyramid and let $G$ be its centroid. A$\frac {MA} {MV}+ \frac {NB} {NV}+ \frac {PC} {PV}=1.$ I tried to use Menelaus' theorem, but I didn't get to any result.
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1answer
79 views

Euclidean Inner Product in R^n

A matrix defined inner product of R^n generated by the invertible nxn matrix A, < u,v >= Au dot Av. An orthogonal matrix is an invertible matrix where A^T=A^-1 The question asks to prove that if A ...
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Hard Euclidean Geometry question

Let $I$ be the incenter of a triangle $ABC$ and $M$ be the mid-point of the side $BC$. If the line $IM$ cuts the height $AH$ in the point $E$, show that $AE=r$, where $r$ is the radius of the circle ...
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Scalar directions other than an angle

I can think of two obvious, simple ways to express a direction on the two-dimensional plane: as an angle, or as a direction vector (normalized or not). When doing computational geometry, direction ...
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1answer
22 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 $u\cdot v=(u_1^*\cdot v_1, u_2^*\cdot v_2,\cdots\cdots, u_n^*\cdot v_n)$ ...
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1answer
34 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 ...
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55 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 ...
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60 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 \...
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1answer
65 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: Let $...
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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 ...
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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 that ...
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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 ...
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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, ...
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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 ...
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2answers
57 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 = ...
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14 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 ...
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2answers
53 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. :...
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1answer
71 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 ...
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62 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 ...
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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 ...
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1answer
42 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 $...
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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 $A_1$ ...
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1answer
61 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 ...
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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 v\in\mathbb{R}^{n}.\left\|A^kv\right\|_p\!\!=\left\|v\...
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1answer
57 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 <...
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1answer
43 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: ...
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2answers
51 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 ...
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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 ...
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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}} \...