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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172 views

Faster Distance formulae for higher n dimension

I need to calculate the distance two points, but these two points contain more than 100 dimensions. With the regular two or three dimension distance formula, we can extend this further to n dimension ...
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2answers
66 views

Determine whether the triangles $ABC$ and $DEF$ are rectangles

How can we determine whether the triangles $ABC$ and $DEF$ are rectangles? We have $A(-6,5),B(-3,3),C(1,9),D(1,3),E(5,1),F(11,10)$.
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2answers
151 views

finding area of the fourth circle

Three circles of the same radius are arranged in such way that one circle is tangent to the other two. A fourth circle is drawn so that it will contain three circles and be tangent to the other ...
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2answers
506 views

A geometry problem from maths competition

I have a problem understanding the proof. Given an acute angle $A$. Choose an arbitrary point $P$ from the bisector of $A$ and another point $B$ from the side of angle $A$. Draw a line $l$ going ...
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1answer
970 views

Find volume of crossed cylinders without calculus.

I found this puzzle here. (It's labeled "crossed cylinders".) Here's the description: Two cylinders of equal radius are intersected at right angles as shown at left. Find the volume of the ...
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2answers
135 views

area of a circle - 3/4th

How to find the pixels of that line which is crossing the circle? Is there any formula? Iam getting the line's end points
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1answer
86 views

Euclidean Conjugation group

Having a tad bit trying to prove this question, Show that the set of reflections is a complete conjugacy class in the euclidean group E. Also, do the same for the set of half-turns and inversions. ...
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1answer
254 views

Who first discovered that the torus supports a flat structure?

Who first recognized that there exists a homogenous metric on the closed genus 1 orientable surface?
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1answer
43 views

Determining the vectors of a tetrahedron

Suppose that a regular tetrahedron with vertices $A$, $B$, $C$, and $D$ has its centroid at the origin $O$, as in the below schematic. Vectors $OA$, $OB$, $OC$, and $OD$ each have length $\ell$ ($|OA| ...
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4answers
4k views

Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1

How would I go about proving this: Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1
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0answers
257 views

Calculate coordinates of the a point in space with hypotenuse and two angles given

I have a cylinder with a length of $2$, and two angles for rotation around two of the axes. Functions for that are named $\text{RotX}$ (rotation around X axis) and $\text{RotZ}$ (rotation around Z ...
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2answers
83 views

Maximal area of triangle if angle and opposite side length is known

Two lines $l_1$ and $l_2$ intersects at point $A$ such that the angle they intersect is $\alpha$. A line segment has endpoints $B$ and $C$ in the lines $l_1$ and $l_2$, respectively, and $|BC|=l$. ...
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1answer
228 views

Maximum possible area of triangle PQR

A point $P$ is given on the circumference of a circle of radius $r$.Chords $QR$ are drawn parallel to the tangent at $P$.Then how can we determine the largest possible area of triangle $PQR$? Thanks. ...
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2answers
173 views

Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle

How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
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1answer
135 views

there are two docks

There are two docks, dock A and Dock B, on a large lake. The distance between the two docks is 72.5 km. Dock B is directly east of dock A. One day, a steam boat leaves from dock A at noon, and heads ...
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3answers
166 views

Construct tangent to a circle

Using a ruler and a compass how can construct a line through a point and tangent to a circle. What I don't want is to eyeball the line by trying to line-up the ruler over the circle. Best if I could ...
2
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1answer
457 views

Prove that the Simson line of $P$ bisects the segment $HP$ from the orthocentre $H$ to $P$

Let $ABC$ be a triangle with orthocentre $H$ and circumcircle $\odot(ABC)$. Suppose $P\in\odot(ABC)$. Let $\gamma$ be Simson's line of $P$ wrt $ABC$. Prove that $\gamma$ bisects $PH$.
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1answer
603 views

Relationship between the sides of inscribed polygons

In my math textbook there's a demonstration for the calculus of the circumference of a circle that involves regular polygons inscribed in the circle, but I don't get it. The book gives the following ...
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2answers
29 views

Possible constraints on weight constraints for a point position

Suppose a line is given in 2D, and a set of $k$ arbitrary points $x_1, x_2, \dots, x_k$ along that line. For some non-zero weights $w_i$ associated with each $x_1$, it can be shown that a weighted ...
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2answers
740 views

Center of Mass of Quadrilateral

I recently started studying Mass Points and the question arose: If you have a quadrilateral with a mass of 1 at each vertex, how do you locate the center of mass. I had several approaches but I was ...
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6answers
783 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
3
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2answers
121 views

Is every triangle a quadrilateral?

I can imagine a quadrilateral where one of the angles is $180^\circ$. Is this still considered a quadrilateral? More generally, is every $n$-gon also a $(n+1)$-gon (for $n \ge 3$)?
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1answer
6k views

Can two planes intersect in a point?

Is it true that two planes may intersect in a point ? or If they intersect then, they always make a straight line ? I have some doubt; please explain.
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0answers
235 views

Menelaus's Theorem clarification

If three points, one on each side of a triangle (extended if necessary) are collinear, then the product of the ratios of division of the sides by the points is -1 if internal ratios are considered ...
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0answers
269 views

What is requirement to inscibe sphere in pyramid with quadrilateral base?

In certain math problem the only information about pyramid with quadrilateral base is that you can inscribe sphere inside. What kind of constraints was put on my pyramid ABCDS? Is it something ...
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3answers
188 views

Sides of triangle and an altitude

Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Let $h$ be the altitude drawn on the side of length $a$ Then is $a^2 + 4h^2 - (b+c)^2$ always negative ?
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2answers
196 views

Reading geometry problems.

Prove that the external bisector of an angle of a triangle (not isosceles) divides the opposite side (externally) into two segments proportional to the sides of the triangle adjacent to the angle ...
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1answer
112 views

Is this circular logic on geometry proof?

I am trying to prove that the internal bisectors of the angles of a triangle meet at a point - the incenter. I need someone to critique this incomplete proof for me Consider $\triangle ABC$ ...
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0answers
26 views

3D orientations from distance constraints

I want to determine the relative orientations within a set of rigid 3D objects given some pairwise distances between certain points on pairs of objects. There are sufficient constrains to fully ...
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1answer
44 views

Reason for hardness of optimal minimization, and use of iterative optimizers

Suppose a set of $n-1$ are given in 2D space, $x_1, x_2, \dots, x_{n-1}$, and an additional point $x_n$ is to be assigned a 2D coordinate such that the prescribed Euclidean distances $d_1, d_2, \dots, ...
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3answers
167 views

Number of ellipses through two fixed points in 2D space?

How many ellipses with a given size (mean $a$ and $b$ given) one can draw through two fixed points in 2D plane?
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2answers
354 views

Hidden geometrical gems in Euclid's Elements?

I am teaching a course in Euclidean geometry at the University of South Carolina, and it seemed highly appropriate and interesting to read Euclid himself. (See here for a wonderful, and completely ...
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2answers
262 views

line equidistant from two sets in the plane

Suppose $A$ and $B$ are disjoint subsets of the plane, both closed, nonempty, and connected. Define $E(A, B)$ as the set of points in the plane equidistant from $A$ and $B$. For example, if $A$ is a ...
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1answer
134 views

Proving $R\ge 2 r$ using synthetic geometry

If $R$ and $r$ be the radii of the circumcircle and incircle of a triangle, then how do I prove by synthetic geometry(i.e. without trigonometry) that $R\ge 2r$? I am aware of a trigonometric proof ...
3
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1answer
2k views

Average Distance Between Random Points in a Rectangle

My question is similar to this one but for rectangles instead of lines. Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed ...
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0answers
38 views

Euclidian embedding of lines

I'm looking for a way to convert a set of lines in R^3 into points in R^n so that distance between any pair of points points is a good approximation of the distance between corresponding pair of ...
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1answer
640 views

Intersection of Two Circles

I have two circles as: $C_1: (x-x_1)^2+(y-y_1)^2=r_1^2$ and $C_2: (x-x_2)^2+(y-y_2)^2 =r_2^2$ and these circles have non-empty intersection. In other words $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\leq ...
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2answers
484 views

Similarity involving Miquel's Theorem

Let $\Delta ABC$ be a triangle. If we place points $D,\ E,\ F$ arbitrarily on the sides $\overline{AB},\ \overline{BC}$ and $\overline{CA}$ respectively, then the circumcircles of the triangles ...
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1answer
611 views

Relation: average (squared) distance to all points, and (squared) distance to centroid

Suppose a set of $n$ high-dimensional points is given. It is known that the sum of all pair-wise squared Euclidean distances is proportional to sum of squared distances of all points to the centroid. ...
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2answers
541 views

Rectangular problem

I was trying to solve this problem: Let P be a point in the interior of rectangle ABCD. Given PA = 3, PD = 4 and PC = 5, find PB. I feel lost because it's not right to assume P is in the center ...
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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 ...
0
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1answer
109 views

Finding the x- coordinate in triangle

Is it possible to find the point which is marked by question mark ? we know that the s1(x)=s2(x) (the areas of the two triangles are equal)
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2answers
494 views

Cutting the corners of a cube

Do you know of any way to cut the corners of a cube by means of rotation assuming that the cube is centered in the origin of XoYZ? For example if we have a square centered in the origin of XoY and we ...
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2answers
223 views

Intersection of a unit sphere of a given norm in finite dimension with an hyperplane.

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$. Let $C:=\{x\in\mathbb{R}^n\,:\,\|x\| \leq 1\}$, that is to say let $C$ be a convex compact symmetric set of non empty interior. Let $H$ be a linear ...
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1answer
79 views

Locus perpendicular to a plane in $\mathcal{R}^4$

I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me.. In $\mathcal{R}^4$ I have a plane $\pi$ and a point P. I have to find the ...
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1answer
666 views

find the center of an ellipse given tangent point and angle

I have an ellipse with known major radius $r_x$ and minor radius $r_y$, aligned with the x- and y-axis. Given a tangent point $T$ and the tangent angle $\alpha$, how do I calculate the center $C$ ...
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1answer
68 views

find point on the line in $R^n$

I am trying to find the coordinates of a middle point of a line in $\mathbb{R}^n$. Let $X(x_1, \ldots, x_n)$ and $Y=(y_1, \ldots, y_n)$ be two points in $\mathbb{R}^n$. How do I find the middle ...
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2answers
2k views

Solid angle between vectors in n-dimensional space

There is a formula of to calculate the angle between two normalized vectors: $$\alpha=\arccos \frac {\vec{a} \cdot\ \vec{b}} {|\vec {a}||\vec {b}|}.$$ The formula of 3D solid angle between three ...
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0answers
92 views

What property does this equation calculate?

It's pretty difficult to Google for the meaning of a formula. This is the equation, it has to do with ellipses and GIS coordinates. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
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1answer
118 views

Equivalence to the Euclidean Parallel Postulate

Show that the proposition P : There exists a pair of straight lines that are at constant distance from each other. is equivalent to the Parallel Postulate Q : If two lines are drawn which ...