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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Are Euclidean distances a monotone function of inner products?

Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of $k$ vectors happens to decrease? Similarly-if decrease is ...
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2answers
26 views

Orientation of vector relative to other vector

Given two directional vectors in 2D space, $\vec v=(v_x, v_y)$ and $\vec w=(w_x, w_y)$, what is the easiest way to calculate if $\vec w$ is orientated clockwise or counterclockwise relative to $\vec ...
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1answer
40 views

Tangents to a circle

For this construction, how would you show that the perimeter of the triangle $CDF$ is equal to $2BC$? Please include steps and whatnot.
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47 views

Nature of Points and Lines in Euclidean Geometry

It may be true that very few middle school student can grasp the meaning of lines and points in Euclidean geometry prior to a direct instruction. For example, it's possible that such a conversation ...
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1answer
46 views

volume of the solid

Using geometry, calculate the volume of the solid under $z = \sqrt{49- x^2- y^2}$ and over the circular disk $x^2+ y^2\leq49$. I am really confused for finding the limits of integration. Any help?
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257 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
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188 views

Vertex-transitive polytope with large facet

Consider a vertex-transitive convex polytope with a facet containing more than the half of all vertices. Does it already have to be a simplex or are there other examples? I am particularly interested ...
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3answers
89 views

Proof that a line cuts in half the area of a parallelogram iff it goes through the intersection of the diagonals?

I read a theorem in a book which says that a line bisects a parallelogram iff it goes through the intersection of the diagonals. The edge case of this is of course if the line is one of the diagonal ...
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3answers
245 views

Rationale for a convention: Why use the semiperimeter in Heron's formula?

Heron's formula says that the area of a triangle whose sides have lengths $a, b, c$ is $\sqrt{s(s-a)(s-b)(s-c)}$ where $s=(a+b+c)/2$ is the semiperimeter. It can also be stated by saying that the ...
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0answers
31 views

Are a half-disk and a quarter-disk scissors-congruent?

Call two measurable subsets of the plane of equal area scissors-congruent if one can be decomposed into a finite number of measurable pieces which may be reassembled to yield the second. Any polygon ...
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1answer
201 views

Reducing the maximum euclidean distance

This question comes from the HackerRank's "20/20 Hack February" contest which has now ended (problem link). There are N bikers present in a city (shaped as a grid) having M bikes. All the bikers ...
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2answers
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The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
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617 views

Why are there so few Euclidean geometry problems that remain unsolved?

Stillwell mentions in his book Mathematics and its History that: Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers... What is it ...
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0answers
74 views

Conic equation from cone/plane intersection

In an orthonormal cartesian frame $(O; \vec{x}, \vec{y}, \vec{z})$ consider: an infinite plane $P$ defined by: a point $p = (p_x, p_y, pz)$ an normal vector $\vec{n} = (n_x, n_y, n_z)$ a cone $C$ ...
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1answer
40 views

Geometrically prove that for a point on a diameter…

Geometrically prove that for a point on a diameter between the center point and the perimeter of a circle, the distance between this non-center point is the shortest distance to the perimeter. So $A$ ...
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1answer
53 views

Geometry question considering triangles and cyclic quads

Let $\triangle ABC$ be a triangle and $P$ a point on the circumcircle of this triangle. Let $U$, $V$, and $W$ be the projections of $P$ onto the three sides of the triangle. Show that the points $U$, ...
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1answer
90 views

Area of triangle inside triangle

In triangle $ABC$ we choose 3 points $D,E,F$, such that $\overline{AD} = \frac 13 \overline{AB}, \overline{BE} = \frac 13 \overline{BC}, \overline{CF} = \frac 13 \overline{CA}$. Draw segments ...
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1answer
117 views

Triangle inequality for an obtuse triangle

$\alpha < 45^\circ$, how to show that 1) $|AB+AC|>|DB+DC|$? 2) $|AB+AC|>|DB+DC+DA|$?
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34 views

Which kinds of geometry have an angle measure?

It is possible to construct alternative planes to the usual Euclidean plane $\mathbb{R}^2$ by replacing the real numbers $\mathbb{R}$ with other ordered fields $F$. Depending on the choice of $F$, ...
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1answer
52 views

Linear distance is proportional to angular distance, why?

Im my Fourier series book, the following is stated: We may specify the position of a point on the circle by its angular coordinate $\theta$, measured from some fixed base point. Since linear distance ...
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1answer
40 views

Determine missing angle in polygon

I'm trying to figure out this question: Determine the measure of angle a I'm guessing $a=96\unicode{0186}$ using the following work: $$a = 180 - 84 = 96 $$ ...
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2answers
360 views

Gödel's (in)completeness theorems and the axiomatization of Euclidean geometry

In David Hilbert's 1899 Grundlagen der Geometrie, Hilbert gives a rigorous axiomatization of Euclidean geometry. As I understand it, some of Hilbert's axioms must be expressed in second order logic ...
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1answer
53 views

Regular polygon diagonal lengths

Suppose that a regular $n$-gon has integer side length $m$. Is the lengths of its diagonals always algebraic numbers? If yes and if $n,m$ are given, is there an easy way to compute the diagonal ...
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2answers
2k views

What is a straight line?

I have researched this question for days and can not locate a good answer. It could be a mathematical object that is defined by an axiom as Euclid or Hilbert. But if a curve is drawn between two ...
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1answer
55 views

Hermite Normal Form and Reduced Row Echelon form.

After reading about the Hermite Normal form and row echelon form, I find it that both these forms are similar in every respect. My question is, are they similar? Or is Hermite Normal form a special ...
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1answer
36 views

what is the X coordinate of point with Θ-90 degree [closed]

what is the X coordinate of point with Θ-90 degree ? 1- 0 2- 1.414 3- 7.07 4- infinity if I used this x = r sin(Θ)cos(ϕ) how can start I mean that if I substitution with Θ in sin with Θ-90 ...
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1answer
18 views

what is the X component of the point , where the spherical coordinates of point are (100,30,60)?

The spherical coordinates of point are $(100,30,60)$, what is the X component of the point $30$ $43.3$ $50$ $75$ I know that in the spherical coordinates, $$x = r \sin(\theta) \cos(\phi),$$ so ...
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1answer
38 views

spherical coordinates

spherical coordinates of point are $(10,20,30)$, the distance between the point and the origin of coordinate system is ? 1- $10$ 2- $14.4$ 3- $20$ 4- $30$ I know that the distance between two ...
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1answer
27 views

Prove that orthogonality in Euclidean space is geometrically perpendicularity?

Is this simply true by definition (that is, taken as axioms?) How would one to prove that for $||\vec{x}||=1$ and $||\vec{y}||=1$, if $(\vec{x},\vec{y})=0$, then $\vec{x}\perp\vec{y}$? In other ...
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51 views

What do the Purser's theorem says?

Mathworld's entry on Purser's Theorem says the following: Let $t, u$, and $v$ be the lengths of the tangents to a circle $C$ from the vertices of a triangle with sides of lengths $a, b$, and $c$. ...
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2answers
50 views

Bisectors of adjacent angles of a parallelogram meet on midline?

Suppose $KLMN$ is a parallelogram, and that the bisectors of angle $K$ and angle $L$ meet at point $A$. Prove that $A$ is equidistant from $\overline{LM}$ and $\overline{KN}$, without using ...
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1answer
57 views

What are four ways to quadrisect any triangle?

What are four ways to quadrisect any triangle with compass and straightedge? I have a few already: Draw a median and from the midpoint, draw two medians to the remaining sides. Draw a median and ...
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3answers
304 views

What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry?

I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that ...
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1answer
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I am looking for a proof of the “ begonia theorem”.

Let $D$, $E$, $F$ be points on respective (extended) sides $\overleftrightarrow{BC}$, $\overleftrightarrow{CA}$, $\overleftrightarrow{AB}$ of $\triangle ABC$, such that $\overleftrightarrow{AD}$, ...
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1answer
102 views

Find all Pythagorean triples $a<b<c$, where $c=65$.

How can one prove that all the Pythagorean triples satisfying this condition have been found? We are working with positive integers a, b, and c.
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1answer
63 views

there concurrent lines, perpendicular to the sides of a triangle

Given a triangle $\Delta ABC$. Let $A_1,B_1,C_1$ be points on the circum circle of $\Delta ABC$ such that $AA_1\parallel BC , BB_1 \parallel AC, CC_1 \parallel AB$. Through the points $A_1,B_1,C_1$ ...
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1answer
424 views

What are some isometries of $S^2$ without fixed points?

This spherical geometry question involves isometries. I am particularly looking for isometries with no fixed points.
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2answers
150 views

What does the completeness of a system mean for the provability of certain statments?

Through the 16th to 19th centuries mathematicians tried to prove the Euclids parallel postulate from Euclid's other four postulates. In the beginning of the 19th century the mathematics community ...
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1answer
75 views

Pappus Chain Recursive Radii

Here is the equation I am asked to find. I have researched a lot into the Pappus chain, methods primarily involving circle inversion, but can only find examples of calculating the nth radius ...
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67 views

How to determine the length of $x$ in this sketch?

Consider the following construction: Assume that you are given the lengths of the sides $a,b,c,d,e$ as well as the size of the angle $\phi$. The task is to determine the length of $x$ in terms of the ...
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Alternative coordinates for the complex plane $\mathrm{Re}[e^{-is}z]=a$, $\mathrm{Re}[e^{-it}z]=b $

I am defining coordintes on $\mathbb{C}$ using a "generalized" real and imaginary part. Here $a,b \in \mathbb{R}$. \begin{eqnarray*} \mathrm{Re}[e^{-is}z]&=&a \\ ...
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1answer
155 views

How to do you find missing vertix in right triangle in a graph?

![enter preformatted text here][2]I'm graphing a line segment. The end points are $A$ and $B$. Then I"m using $10 \%$ of the length of line segment $AB$ to form ...
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5answers
132 views

Pseudo-pythagorean theorem

Pythagoras' theorem is a special case of the Cosine theorem for a angle of $90°$. But also for an angle of 60° and 120°, "aesthetical" special cases derive: $c^2=a^2+b^2\pm ab$ First question: Are ...
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2answers
268 views

find the equation of the locus of a moving point which is always equidistant from the y-axis and the point (-6,4)

Do you know how to solve its equation? Already solved some locus problems that gives points but not in the y or x axis problems.
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3answers
262 views

How would you prove that the graph of a linear equation is a straight line, and vice versa, at a “high school” level? [duplicate]

This is something I've been wondering about. Namely, I've always accepted "on intuition" that the equation $$ax + by = c$$ is, when graphed, a line. You can plot the points $(x, y)$ satisfying the ...
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1answer
59 views

Basic Euclidean Geometry, Circle Arc

So, here is my problem explained as best as I can. I'm working on some navigation logic for a wheeled vehicle, but I've not the foggiest idea of how to do much path finding, really. So, my basic idea ...
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1answer
68 views

we need to show $Ar(\Delta APD)=Ar(ABCD)$

$ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets $BC$ produced at $P$ we need to show $$Area(\Delta APD)=Area(ABCD)$$ I tried but did not get properly. Thank you for helping.
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77 views

Shining a laser into a mirror maze

I tried to formulate the following problem in a more mechanical way involving soccer balls, but the physics got too unrealistic. I know that what follows could be made more precise, but I hope the ...
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128 views

3D Game: Pitch Yaw Roll of a point

I have a flat elliptical plane and I'm trying to figure out how to represent it based on its direction. So I basically need to calculate its pitch, yaw, and roll. I have a camera at $C$, and a point ...