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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how to derive relation between solid angle and surface area and the radius of sphere using definite integral?

how to derive relation between solid angle and surface area and the radius of sphere ? I know $s=r^2\Omega$ but how they got it using integral ?
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Would it make sense to talk about approximated constructions in euclidean geometry?

I got curious with something: I know that in euclidean geometry we talk about constructible and non-constructible structures, do we have the concept of approximation in euclidean geometry? I mean, we ...
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165 views

Why are the axis labelled as such in the 3d Cartesian coordinate system?

A long time ago I was taught that in 3d space, the x axis is the length/width or left/right space, the y axis is the height, and the z axis is the depth. When we draw things in 2d on a page, this ...
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160 views

Lines $ MF, DE, QR$ in a triangle intersect at one point

In a triangle ABC, a circle is inscribed with center in $I$. The inscribed circle touches sides $BC,CA,AB$ in $D,E,F$ respectively. Join the point $C$ and $F$, $B$ and $E$. Let $Q$ and $R$ be the ...
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552 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
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109 views

How can I estimate the Euclidean distance?

I read in an article the Euclidean distance formula can be estimated with about 6% relative error with the following formula. Would you please why this is true and where can I find such estimations? ...
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1answer
911 views

How to find a point along an vector at a variable distance

I need to find the value along a vector for a given x coordinate. Like so; I know the values of A, B and C. All of these value are variable. I need to calculate X. I know this is possable I just ...
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286 views

How to find the affine transformation(s) ( if any ) that maps one quadrilateral into another.

Given the Fundamental theorem of affine geometry. Let P,Q,R be any three non-collinear points in R2, and let U, V,W be any three other such points. Then there is exactly one affine transformation ...
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47 views

In search of a symmetric homogeneous graph with a pivotal origin

I'm trying to design a computer game and I need a symmetric homogeneous graph with a pivotal origin which will act as the map of the game (players will walk according to it). Here's an example of ...
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2answers
211 views

Circle of Apollonius proof question

Im reading a proof of the Circle of Apollonius and I am unsure of one part of it - Find the locus of a point P whose distances from two fixed points, A and A' are in a ratio of 1 : $\mu$. Define a ...
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1answer
4k views

Does a rhombus bisector creating two equal triangles by reflection imply a square?

A followup to my previous question - Assume we have rhombus wxyz. If I draw a line from w to ...
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Why do we believe the equation $ax+by+c=0$ represents a line?

I'm going for quite a weird question here. As we know, the equation in Cartesian coordinates for a line in 2-dimensional Euclidean geometry is of the form $ax+by+c=0$. I'm wondering why do we ...
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114 views

Calculating the Euclidean Distance

How do I calculate the Euclidean Distance between $(22, 1, 42, 10)$ and $(20, 0, 36, 8)$? I think I know the formula, but I don't know how to apply it to this data. $$d(p, q) = \sqrt{(q_1-p_1)^2 + ...
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1answer
2k views

Finding the length of a side of a quadrilateral given 2 sides and two angles

In the quadrilateral $ABCD$, $AD$ is parallel to $BC$, $\angle C = 2\angle A$, $CD=3$, and $BC=2$, What is $AD$? I think I have to making a line bisecting $AC$, but I am not sure.
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418 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
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4answers
118 views

Prove $|x + y|^2 - |x - y|^2 = 2|x|^2 + 2|y|^2$ and interpret its meaning.

Prove $|x + y|^2 - |x - y|^2 = 2|x|^2 + 2|y|^2$ if $x, y \in \mathbb{R}^k$. Interpret this geometrically, as a statement about parallelograms. I've shown that the expression given equates to $x ...
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1answer
177 views

About intersection points of some diagonals of a regular $n$-gon

Are my expectations true? My expectation 1 : There exist some intersection points, which are not on the center of the regular $n$-gon, of three or more diagonals when you draw all diagonals of a ...
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49 views

Geometry question

In the given figure , AP, BQ and CR are perpendicular to line AC. And AP=$x$ , BQ=$y$, CR=$z$ then find the value of $\frac{1}{x} + \frac{1}{z}$ in terms of $y$. I have no idea how to solve it. But ...
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104 views

Length of Chord is Independent of Point P

This is question 1.49 from Baragar's textbook called A Survey of Classical and Modern Geometries if anybody is familiar with that text. This question is assigned as homework, so I am just looking for ...
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161 views

how do i calculate coordinates of point on rectangle based on angle

If i want to find out what is x and y of point lies of a line drawn from center of the rectangle to outward at certain angle. take a look at picture below.
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How I can show that the point P (Miquel point) is in the circle formed by the centers of the other 4 circles?

This is the theorem If the points $A', B', C'$ on the sides $BC, CA, AB$ of a triangle ABC are collinear, then the centers of the circumcircles of the triangles $\triangle AB'C'$ $\triangle A'BC$ ...
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flat subspace : minimal characterization

In the euclidean space ${\mathbb R}^3$, I can define a plane by three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$, using six reals. Of course I can give an equation $ax+by+cz=d$, using only 4 ...
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Inequalities involving areas of triangles in the plane, and generalizations

It is known that if $O$ is the origin, and $A, B, C, D$ are points in the first quadrant (in $\Re^2$) ordered such that the sequence of slopes of the lines $OA, OB, OC, OD$ is increasing, then $\Delta ...
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518 views

Convert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels $x$ units in Euclidean space, how ...
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417 views

Spiral of Theodorus - Discussion

The fact that $\sqrt2$ is not rational goes back to Theodorus of Cyrene from the school of Pythagoras, and is discussed in Plato's dialog "Theaetetus". Of course, $\sqrt n$ is not rational for any ...
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353 views

Is there a geometrical proof of the impossibility of squaring the circle?

The impossibility of certain constructions in Euclidean geometry, such as squaring the circle with straight-edge and compass is usually shown by using algebraic methods. I am wondering if there are ...
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1answer
492 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
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145 views

How was born the topology from Euclidean geometry?

Good evening, I have the question arose as to create interest starting topology of Euclidean geometry, what was the interest of those who created it? Thanks for your help
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85 views

What is a function that is equidistant from two functions?

For example, let $f(x)$ be $x^2$ and $g(x)$ be $x^3 + 1$. It is easy to see that $h(x)=\frac{f(x)+g(x)}{2}$ is not equidistant from $f$ and $g$. Then, what is a general form of a fucntion $h(x)$ that ...
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Are all metric space as a euclidean space?

I believe that all euclidean space is a metric space? But I need to know about inverse? I mean: are all metric space as a euclidean space? Is there any kind of metric space which is not euclidean ...
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420 views

In triangle abc,angle BAC is 22 degrees. A circle with centre O has AB produced,AC produced and BC tangents. Find the number of degrees in angle BOC.

In triangle abc,angle BAC is 22 degrees. A circle with centre O has AB produced,AC produced and BC tangents. Find the number of degrees in angle BOC. I am not able to interpret it. Plz help me in ...
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174 views

An inconstructible quadrilateral

A student tried to draw a quadrilateral $STOP$ with $ST=5cm$, $TO=4cm$, $\angle S = 20^{\circ}, \angle T = 30^{\circ}, \angle O = 40^{\circ}$. But he found out that it was impossible to construct ...
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309 views

perimeter of square inscribed in the triagle

In the figure given below, PQR is a triangle with sides PQ=10, PR=17, QR=21. ABCD is a square inscribed in the triangle. I want to find perimeter of square ABCD that is to find the length of side AB. ...
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Distance between $3\times3$ matrices (isometries)

The problem is pretty long, so I'm going to describe it while writting what I have so far. First I have to asociate the $3\times3$ matrices with $\Bbb R^9$, so we defined $\phi:M_{3\times3}(\Bbb R) ...
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448 views

how many circle of radius r can be placed inside on the border of another circle

Suppose Radius R of a big circle is given. and I want to place some little circles of radius r inside on the border of that big circle. Like the picture: But how to find how many small circle can ...
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Why only compass and straightedge?

I've read and watched some lectures on euclidean geometry - not so advanced but I've seen the focus on constructions. Two instruments are used, compass and straightedge, I had the following doubts: ...
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Prove that 4 points are concyclic

In $\triangle ABC$ it holds $BC = \frac{AB+AC}{2}$ .Let $M$ and $N$ be midpoints of $AB$ and $AC$ ,and let $I$ be the incentre of $\triangle ABC$. Prove that $A,M,I,N$ are concyclic. I have tried it ...
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169 views

In a convex Quadrilateral ABCD, $\angle ABC = \angle BCD = 120^{\circ}$.Prove that: $ AC + BD \ge AB + BC + CD$

In a convex Quadrilateral ABCD, $\angle ABC = \angle BCD = 120^{\circ}$.Prove that: $$ AC + BD \ge AB + BC + CD$$ My attempt Tried to use cosine formula twice' i.e ($\triangle ABC$ and ...
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39 views

Find the axis of reflection

I have to determine the axis of reflection of the composition of a rotation and a reflection, y show that the order of composition matters. So I multiply the matrices that represent each isometry, ...
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56 views

Simpler solution to a geometry problem

In a set of geometry problems, I got this one: If in a triangle $ABC$ with segments $AB=8$, $BC=4$, and $3A+2B=180^{\circ}$, calculate the side $AC$ My solution was Let ...
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Composition of two reflections is a rotation

I have this problem that says: Prove that in the plane, every rotation about the origin is composition of two reflections in axis on the origin. First I have to say that this is a translation, off my ...
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4answers
270 views

Simple geometry question, to be proved without trigonometry

In triangle $\triangle ABC$, ray $AD$ is a bisector of angle $A$, which intersects $BC$ at $D$. Also given are that $AC$ = 4 cm, $AB$ = 3 cm and $\angle A = 60^\circ$. Find the length of $AD$. This ...
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Solve $10x+2x^2+x^3=20$ using only algebra and geometry?

The cubic formula and modern math is not allowed, only algebra, geometry, and the like. Supposedly this problem was given to Fibonacci. Here is the whole paragraph I read: In Flos Fibonacci gives ...
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Drawing Euclid?

I decided to study Euclid for fun. I have Oliver Bryne's edition. I also want, as much as possible, to construct the figures myself, to get a deeper understanding. How did people traditionally do ...
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Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?

Let's call a point $P$ which satisfies the following condition 'a rational point'. Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ ...
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553 views

What is a the intuition behind a parametric equation?

I have always used equations for the line (y=a + bx) in R2. Recently I came upon this thing called parametric equations. I cannot grasp the difference between them and the equations for lines that I ...
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4answers
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How do I move through an arc between two specific points?

I've found many answers to similar questions here, but I'm still stuck. I want to move an object from point sx,sy to point dx,dy through an arc that bulges by distance b from the line straight ...
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54 views

Geometry finding area problem

A regular 2N -sided polygon of perimeter L has its vertices lying on a circle. Find the radius of the circle and the area of the polygon.
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About the relation between a tetrahedra and spheres moving in a tetrahedra

I found the following question in a book: There exists a regular triangle $OAB$ which has edge-length $2$. Let $H, I, J$ be a foot of the perpendicular line drawn from a point $P$ in $OAB$ to the ...
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512 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 ...