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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Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...
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2answers
112 views

What is the size of the angle $\angle AMC$? [duplicate]

Suppose we have a triangle $\triangle ABC$ where the size of two angles are given: $\angle B=15^\circ$ and $\angle C=30^\circ$. We draw the median $AM$, so now what is the size of angle $\angle AMC$? ...
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1answer
28 views

Is a shape 'polarizable'?

Given a point $p$ inside a shape $S$ described as an $n$-vertex polygon, let us say that $S$ is polar with respect to $p$ if S can be described by a polar equation $r(\theta)$ with $p$ as the origin. ...
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1answer
38 views

Can circumscribing a circle around a polygon prove that the sum of the interior angles of an n-sided polygon is $180(n - 2)$?

I am trying to create my own proof that the sum of the interior angles in a regular polygon is $180(n - 2)$, where $n$ is the number of sides in the polygon. I have seen these proofs for this formula, ...
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1answer
31 views

Victoria Jones want to construct a time capsule. The capsule will be right circular cylinder

Victoria Jones want to construct a time capsule. The capsule will be right circular cylinder of height 'h' cm, and radius 'r' cm on each end . Let the total volume of capsule be V cm^3. Express V in ...
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Convex quadrilateral with interior point

Let $ABCD$ be a convex quadrilateral and $X$ is an interior point. Also let $AX\cap BD=\{E\}$, $BX\cap AC=\{F\}$, $CX\cap BD=\{G\}$ and $DX\cap AC=\{H\}$. Prove that: $$AF\cdot BG\cdot CH\cdot DE=...
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6answers
648 views

Given distances (shortest paths) between four cities, how to show that they cannot be in the same plane?

In the example below we are given distances between four cities. The author of the book says that these distances "suffice to prove that the world is not flat". Do I understand this correctly that ...
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0answers
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What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
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2answers
30 views

Draw polynomials to demonstrate Euclid's axioms.

I've a problem with Euclid's axioms. I understand them, but now I want some equations (polynomials) that I can use to draw some graphics and probe these axioms. For example, a rect equation that ...
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1answer
47 views

Geometric proof of expansions of series

I have read that Barrow had proved the fundamental theorem of calculus. I have read that proof and its a good. Further I know Newton had derived the sine and cosine series. His methods obviously didn'...
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1answer
35 views

Finding points on a right triangle

I have the points A, B and C. I also have the angle alpha between AB and BD or BE, and I know l = |BD| or |BE|. But how can I find D or E?
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173 views

A challenging straightedge and compass construction

Three points $A,O,B$ are given, and $0<\theta=\widehat{AOB}<\frac{\pi}{3}$. It is known that there are two points $A',B'$ on the segments $OA,OB$ such that $$ BB'=B'A'=A'A $$ holds. How ...
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0answers
29 views

how to find the angle of Lovasz umbrella

in the book Thirty-three Miniatures: Mathematical and Algorithmic Applications of in problem 28 The Secret Agent and the Umbrella page 132 (pdf 140) we want to find an orthogonal reperesentation of ...
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2answers
169 views

Triangle in perspective to a given triangle but similar to another

Is it always possible to construct a triangle that is in perspective to a given triangle and have it also be similar to a different given triangle? If you create a triangle in perspective to another, ...
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2answers
25 views

Prove that $\frac{AB}{PB}\cdot \frac{BQ}{QC}\cdot \frac{CR}{RD}\cdot\frac{DS}{SA}= 1$

If the sides AB, BC, CD and DA of a quadrilateral ABCD are cut by straight lines at points P, Q, R and S respectively, how do I prove that $\frac{AB}{PB}\cdot \frac{BQ}{QC}\cdot \frac{CR}{RD}\cdot\...
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2answers
127 views

Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel? [closed]

Below please enjoy a Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel? The below construction is created by beginning ...
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1answer
59 views

Prove that an isometry preserves straight lines using intuitive geometry

This is an exercise of the book "Basic Mathematics" by Serge Lang, p.145. I've been working on this proof for a few days and I can't seem to make it more coherent than this. Would appreciate some help....
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36 views

When does there exist a point with a given ratio of distances to the vertices of a triangle?

I have the triangle ABC and an unknown point P not necessarily inside the triangle. Also, I have three lengths (...
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1answer
40 views

Compass only constructions

Solve this using only a compass. Beyond stumped... please help.
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3answers
257 views

about a ninth-grade geometry problem

My brother asked me this problem, and he is studying ninth-grade. I can't solve it using primitive tools of pure geometry. Hope someone can give me a hint to solve it. Thanks. Given a circle $(O, ...
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1answer
33 views

Prove that $OP^2=PC^2+OB^2$.

Suppose you have A,B,C,D four points in harmonic range and O, P are the midpoints of AB and CD respectively. Prove that $OP^2=PC^2+OB^2$. I would guess that this is a sort of corollary to the theorem ...
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1answer
16 views

Shape of polar set

Let $K$ be a subset of $R^n$, which contains the origin $\theta$ , maybe , it is needed that it is not very strange . The polar set of $K$ is $$ K^0=\{x\in R^n : \langle x,y \rangle \le 1 ~~~\forall ...
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9answers
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How do curves consist of points?

According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in ...
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2answers
48 views

Volume of intersection of the $n$-ball with a hyperplane

Let $\mathcal{B}_n$ be the $n$-ball of radius $r>0$ and centre $\mathbf{x}_0$, i.e., $\mathcal{B}_n=\{\mathbf{x}\in\mathbb{R}^n\colon \|\mathbf{x}-\mathbf{x}_0\| \leq r\}$. The volume of $\...
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321 views

A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio construction with square and circle?

Note: this construction is a vastly expanded version of my earlier construction here: Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or ...
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1answer
66 views

New Golden Ratio Conjecture with Triangle and Square: It is very close, but is it really the golden ratio?

Geogebra gives me 1.616 for the ratio of the blue segment p to the red segment q instead of the golden ratio 1.618 for the construction shown below, so it could be close to PHI, but no cigar. This ...
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1answer
54 views

Inverse points concurrence on the circumcircle

Let $ABC$ be a triangle, and $P,Q$ two inverse points with respect to its circumcircle. Let the circle through $A,P,Q$ meet $AB,AC$ at $A_c,A_b$ respectively. Analogously define $B_a,B_c,C_a,C_b$. Let ...
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2answers
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placing balls inside ball [duplicate]

Is it possible to put pairwise disjoint open 3d-balls with radii $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ inside a unit ball? not an original question, I found it somewhere in the internet once,...
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1answer
29 views

Calculus resultant and equilibrant question

Two forces of 40 N and 50 N act at an angle 60 degrees of to each other. Determine the resultant and equilibrant of these forces. Let vector u = 40N Let vector v = 50N I made a right angle triangle ...
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2answers
49 views

Geometry construction

I appreciate any help. I want to find the angle $ADC$. I have drawn the circle in Geogebra, and the angle $ADC=120^\circ.$ But how can I give an argument that is always will be $120^\circ$ if angle $...
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3answers
89 views

New very simple Golden Number Ratio PHI construction with Circle and Two Equal Segments of Circle Diameter. Is there prior art? Proofs?

Geogebra gives me the golden number PHI to fifteen decimal places for this simple construction illustrated below wherein the ratio of the blue line i to the red line h is PHI or 1.6180.... The golden ...
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0answers
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I am going to work in SE(3) group, is vector sum approach applicable in this group?

I am working on control of mobile robot in 3d. I want to do vector sum for X and Y components, use this vector sum in control methodology and again convert resulting speeds and torques into their X ...
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2answers
35 views

Equation for a circle in homogeneous coordinates

The equation for a circle in homogeneous coordinates is given by $(x - aw)^2 + (y - bw)^2 = r^2w^2$. I understand that the center of the circle, given by (a, b) in euclidian space is given by (a, b, ...
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New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length. Is there any prior art?

Consider three regular polygons with 3, 4, and 5 sides wherein all the polygons have sides of equal length X throughout, as illustrated below. The ratio of the red line segment a to the blue line ...
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1answer
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Geometric Significance that 2D Points Form a Line

I'm reading through Multiple View Geometry in Computer Vision, by Hartley and Zisserman, and on page 2 it is stated that points at infinity in the two-dimensional projective space form a line, and in ...
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3answers
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New, extremely simple golden ratio construction with two identical circles and line. Is there any prior art? [duplicate]

This question is different from a previously asked question (linked above) as this golden ratio construction only utilizes two circles and a line, and is thus far simpler than the golden ratio ...
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0answers
31 views

How to find the Coordinate equation of a curve which bends all the parallel rays from infinity towards a single point

How should I proceed on to find the coordinate equation of a curve such that it bends all the parallel rays coming from infinity towards a single point. Yes I know that it would be a 2nd degree ...
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2answers
152 views

New Golden Ratio Construction with Two Adjacent Squares and Circle. Have you seen anything similar?

The below Golden Ratio Construction results in a ratio of PHI (1.6180...) between the blue line and red line, as found in Geogebra. This seems like a simple construction of the golden ratio, yet so ...
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1answer
110 views

Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or 1.6180.. exactly

Note this golden ratio construction has been dramatically updated here with numerous golden harmonies: A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio ...
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3answers
43 views

Geometry Question Help

The bases of trapezoid $ABCD$ are $\overline{AB}$ and $\overline{CD}$. We are given that $CD = 8$, $AD = BC = 7$, and $BD = 9$. Find the area of the trapezoid.
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Finding a point coordinate given some distance restrictions relative to other points

I want to find the solution space of coordinates for point $p$ that satisfies the following system: $$ \begin{cases} [distance(p,a) - distance(p,b)] = k_1\\ [distance(p,c) - distance(p,d)] = k_2 \end{...
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2answers
70 views

New Golden Ratio Construct with Geogebra using Square and Triangle with Same Base Width. Geometric proof of golden section?

The below construct of the golden ratio, based on the ratio of segment c to segemnt b, is so very close to PHI. Geogebra gives the value of 1.61957 instead of 1.61803. Might anyone have any insight ...
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2answers
27 views

mod used to describe an angle

Reading Pedoe's "Geometry: A Comprehensiveness Course" I came across the following We know that from Euclidean geometry, for any triangle ABC,$$\sphericalangle ABC + \sphericalangle CAB + \...
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1answer
30 views

Scalar product is 0 in any triangle

How can we prove that the following scalar product relation holds in any triangle? $$\left [-\overrightarrow{AB}\tan B (\tan A +2\tan C)+\overrightarrow{AC}\tan C (\tan A+2\tan B)\right ]\cdot \left (...
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0answers
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There is a square that vertices are (0,0) (0,2) (2,0) (2,2) [duplicate]

A point P satisfies following condition : The straight line passing through P and dividing the area of the square by 1:3 does not exist. Can we know the locus of P and the area of the locus ? I ...
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2answers
98 views

How many mutually orthogonal circles are possible?

How many mutually orthogonal circles is it possible to have? It is easy to construct $3$ mutually orthogonal circles, e.g. $3$ circles with radius $1$ and centers at the vertices of an equilateral ...
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1answer
23 views

Problems on measure of angles and arcs in a circle diagram

A friend of mine recommended this site. I cannot figure out any of the parts in the problem in the picture click here The line segments AE and DE are not tangent to the circle, so I don't know where ...
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0answers
25 views

Does 3D euclidean space allows vector sum in 2 dimensions?

Is this right to add two orthogonal vectors to to get one vector, using this vector in calculations and after getting results, decomposing result vector to get orthogonal components? I am a programmer,...
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1answer
48 views

Formula for area of triangle in complex plane [closed]

If $A(z_1)$, $B(z_2)$, $C(z_3)$ are vertices of a triangle $ABC$ in Argand plane, what is the area of the triangle?
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There is a square $Q$ consisting of $(0,0), (2,0), (0,2), (2,2)$

There is a square $Q$ consisting of $(0,0), (2,0), (0,2), (2,2)$. A point $P$ satisfies following condition: The straight line passing through $P$ and dividing the area of square $Q$ in the ...