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
35 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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1answer
34 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 ...
9
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3answers
223 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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2answers
181 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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1answer
29 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
17 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
35 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
23 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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0answers
43 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
39 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
44 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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1answer
195 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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1answer
69 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
48 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
95 views

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 $\overleftrightarrow{AD}$, ...
2
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1answer
65 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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0answers
20 views

When the euclidean distance criteria does not work?

Good day to everyone. I have a question about classification in presence of additive gaussian noise. Assuming a predefined reference set of $N$ complex-valued vectors ...
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1answer
48 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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1answer
112 views

Maximizing the perimeter of a triangle inside a square

BdMO 2014: We have a square $ABCD$ of side length 5.We take a point $E$ on $AD$ and $F$ on $AB$ so that $\angle FCE=45^\circ$. What can be the maximum perimeter of $\triangle AEF$? I can ...
3
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0answers
62 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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2answers
114 views

The angle $168^\circ$ is constructible

Prove that the angle $\theta=168^\circ$ is constructible using a straightedge and a compass. It is enough to show that the number $\cos\theta$ is constructible, and WolframAlpha gave ...
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0answers
15 views

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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2answers
48 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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1answer
119 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
121 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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3answers
105 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
42 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
65 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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61 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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0answers
254 views

Counting simple quadrilaterals in a rectangular lattice.

I've been trying to make an algorithm to find the number of all possible simple quadrilaterals in a N*M lattice. I already have a brute force solution but since this is a Project Euler problem I ...
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1answer
54 views

Pitch, roll, yaw rotations?

I have a random orientation in a room described by pitch, yaw, and roll angles. When I do ...
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1answer
590 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
93 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 ...
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1answer
35 views

Given a triangle $ABC$, with altitude $AD$ and circumcircle radius $R$, show that $AD = 2R\sin\ B\sin\ C$.

Given a triangle $ABC$, with altitude $AD$ and circumcircle radius $R$, show that $$AD = 2R\sin\ B\sin\ C.$$ I'm a bit stumped as to how the altitude of $ABC$ and the circumcircle radius interact ...
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1answer
67 views

Find the shaded area

Find the shaded area Here is the equation that i've made \begin{align*} S&=\pi R^2\\ S_1&=\pi {R_1}^2\left(\frac{24}{360}\right)\\ S_2&=\pi{R_2}^2\left(\frac{24}{360}\right) \end{align*} ...
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2answers
61 views

Geometry problem about angles and triangles

I've been working on this problem for a while. It doesn't seem to hard, but I cannot reach a satisfying solution. The triangle $ABC$ is isosceles with base $\overline{AC}$. A point $O$ is also ...
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1answer
28 views

How do I proof that $\angle ABP =\angle AP'B$ and that $P$, $Q$, $Q'$ and $P'$ are on 1 circle?

Given is a circle with center $M$ and a diameter $AB$. $k$ is the tangent to the circle at point $B$. On the circle there are two points called $P$ and $Q$, such that $P$ and $Q$ are both on the same ...
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2answers
97 views

How do I prove that $CP > \frac 1 2 (AC+BC-AB)$? [closed]

Given is the triangle $ABC$ with point $P$ on side $AB$. How do I prove that $$CP > \frac 1 2 (AC+BC-AB)?$$
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1answer
72 views

A problem of forming equal angles in plane geometry

C and D are two points on the same side of a straight line AB. Find a point X on AB such that angles CXA and DXB are equal.
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2answers
214 views

Formula to find the third point of triangle when two points and all sides are known?

I am writing a program in java. I looking for formula to determine the 3rd point in a triangle if the length of all sides and the coordinates of two points are known.
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2answers
65 views

If $ABCD$ is a cyclic quadrilateral, then $AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD)$

If $ABCD$ is a cyclic quadrilateral, then $$ AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD) $$ I tried using many approaches, but I could not find a proper solution. Can anyone please ...
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1answer
66 views

Quadrilateral problem

Assume a quadrilateral $ABCD$ and $M, N$ points on $AB$ and $CD$ respectively, such as $\frac{AM}{MB}=\frac{CN}{ND}$. Lines $AN$ and $MD$ intersect on $K$ and lines $MC$ and $BN$ intersect on $L$. ...
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0answers
123 views

Area of a equilateral triangle given distances of a point in the triangle from the vertices [closed]

A point $D$ inside an equilateral triangle $PQR$. $D$ is located at a distance of $3$cm, $4$cm and $5$cm respectively from $P$, $Q$ and $R$. What is the area of the triangle $PQR$?
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2answers
106 views

Given a triangle find the length of BC

Given an ABC triangle , if $AB+AD=4$, find the length of BC.
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1answer
105 views

Pythagoras Theorem - Why a Theory? [closed]

Why is Pythagoras Theorem a Theory but not a Law? I mean we use it many times in School and to build stairs etc. and it has been proven however it is called a Theory.
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0answers
39 views

Triangle inequality for angles

For points $O,A,B,C$ in $\mathbb{R}^{3}$, I was trying to show $\angle AOC \le \angle AOB +\angle BOC$. I could show this when all angles were acute. First, I set $O$ to be the origin and $A,C$ to ...
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3answers
130 views

Moscow Math Olympiad 1973

In every polyhedron there is at least one pair of faces with the same number of sides. Solution: Let $N$ be the greatest number of sides in a face of a given polyhedron. Then the number of ...
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0answers
28 views

Area of intersection between square and annulus

The annulus' larger radius is $1$, smaller radius is $r>0$, and center is $(0,0)$. The square's sides are parallel with the axes, the lower left corner's coordinates are $(a,b)$, and the upper ...
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2answers
81 views

How many vectors are needed to define a plane in n dimensions?

How many vectors are needed to define a plane/hyperplane in n-dimensional space? In 3 dimensions, if there are 2 vectors with tails at the origin and the heads in differing locations (and the vectors ...