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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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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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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1answer
55 views

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
20 views

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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1answer
113 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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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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24 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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1answer
42 views

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

Have you seen the attached golden ratio construction before? Three squares (or just two) and circle. For the ratio of segment t to segment a, Geogebra gives PHI or 1.6180.. exactly. Geometric and ...
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0answers
39 views

Faster Alternative than Calculating Euclidian Distance to determine which Coordinate has Max Distance from a fixed coordinate (eg (0,0))

I am developing a program that needs me to determine which coordinate in a $2$-D figure has maximum distance from a fixed coordinate. Let me demonstrate: $3$ points: $(1,3), (2,2), (5,0) $; Fixed ...
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3answers
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Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
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3answers
39 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 ...
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1answer
672 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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2answers
60 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
23 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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4answers
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Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...
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1answer
27 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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1answer
1k views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
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1answer
66 views

Question: What theorem should I use for this geometry problem.

I have already solved this problem using trig, however I feel that their must be an easier way to solve this problem using some theorem or property of quadrilaterals that I am forgetting. Initially ...
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2answers
86 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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3answers
74 views

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 ...
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3answers
863 views

If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?

If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
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0answers
42 views

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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1answer
21 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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1answer
33 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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0answers
24 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 ...
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1answer
35 views

Prove orthogonality

Let $ABC$ be any triangle . Two squares $BAEP$ and $CAFR$ are constructed externally to $ABC$.Let $M$ be the midpoint og $BC$. Show that $AM$ is orthogonal to $EF$. I have no idea how to prove its ...
2
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4answers
75 views

Construct parallel through a triangle satisfying a sum condition

I would like to draw, using the classical compass and rule methods, some points $D$ and $E$ given a triangle $\Delta ABC$ such that $BD + EC = DE$ and $DE$ is parallel to $BC$, as in the following ...
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8answers
19k views

Book recommendation on plane Euclidean geometry

I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
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2answers
29 views

Question about angles of rhoumbus

Problem: Consider a rhombus (Diamond) such that each of its side is the geometric mean of its diameters. I mean if length of each side is X and the diameters a and b; then $X^2$ = a.b Find the ...
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1answer
48 views

Question about area and triangle

Problem: Consider the following diagram. in $\triangle$ABC: Areas: $\triangle$AOM = a $\triangle$POC = b $\triangle$NOC = c $\triangle$BON = d. Find the area of $\triangle$MOB and ...
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1answer
27 views

The sum of distances from the sides of a regular polygon to an interior point is a constant

Let there be a regular polygon of $n$ sides. Assume there is a point $P$ inside the polygon, then prove that $$a_1 + a_2 + a_3 + \cdots + a_n= \text{constant}$$ where $a_i$ is the distance of ...
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1answer
37 views

Geometry Question about Area and surface

Problem. According to following diagram, prove (Area of (MM'N'N)) = 1/3*(Area of ABCD)). We Know that AN = NM = MB and DN' = N'M' = M'C. and quadrilateral ABCD is not and special quadrilateral. ...
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1answer
27 views

Geometry Question About Angles (Triangle) [closed]

Let $\triangle ABC$ be an isosceles triangle ($AB = AC$ and $\angle ABC = \angle ACB = 35^\circ$). We have a point $M$ inside the triangle such that $\angle MBC = 30^\circ$ and $\angle MCB = ...
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2answers
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Find the relation (above/below) a plane and a line

$l: x=0,y=t,z=t$ and $\pi:6x+2y-2z=3$ find if they are parallel and how is above the other. So I took the dot product $(0,1,1)\cdot(6,2,-2)=0$ so they are parallel. To test how is above/below I ...
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1answer
53 views

What is the length of side $AB$, given the following?

It is a Parallelogram. Given that the perimeter is $22$, find $AB$. r I did this it feels wrong. 3x-2 = x-w+1 3-2x = w 3-4w = 3-12+8x 3-4w= 2y+1 3-12+8x=2y+1 2-12+8x=2y 5-4x=y ...
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1answer
1k views

calculating volume of a horizontal cylindrical tank from depth

Has anyone found a good formula to convert a depth of fluid to a volume remaining in a tank for a cylindrical tank laying horizontal? (with or without half sphere end caps) Much Thanks!
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1answer
34 views

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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3answers
42 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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2answers
54 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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1answer
19 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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1answer
32 views

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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3answers
37 views

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 ...
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2answers
28 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
66 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
18 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 ...
2
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1answer
29 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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0answers
14 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 ...
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1answer
23 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 ...
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1answer
26 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 ...