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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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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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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873 views

How do curves consist of points?

According to Euclid, 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 point has no dimensions ie in other ...
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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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69 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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55 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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47 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 $B_a,B_c,C_a,C_b$. ...
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placing balls inside ball

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 ...
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18 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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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
75 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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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
33 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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4answers
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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
22 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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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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29 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
130 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
71 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
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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2answers
63 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
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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
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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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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2answers
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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
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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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 ...
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1answer
34 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 ...
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4answers
76 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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1answer
69 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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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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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
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
38 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
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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
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
46 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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1answer
20 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
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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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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
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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 ...
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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
56 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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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 ...
2
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1answer
31 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
28 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 ...
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1answer
26 views

Minimum Perimeter of a triangle

I have been playing the app Euclidea, I have been doing quite well but this one has me stumped. "Construct a triangle whose perimeter is the minimum possible whose vertices lie on two side of the ...