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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-1
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1answer
216 views

Spherical coordinates to cartesian coordinates.

I want to find out the distance between the centers of $2$ circles. Say, circle $1$ $(\theta,\phi)$ circle $2$ $(\theta,\phi)$ The radius of this circle is found using $d\tan(\theta)$ where $d$ is ...
2
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0answers
123 views

Five squares in a box.

Let $A,B,C,D$ and $E$ be five non-overlapping squares inside a unit square, of side lengths $a,b,c,d,e$, respectively. Prove that $$a+b+c+d+e \le 2.$$ (This problem is a continuation of my previous ...
3
votes
3answers
464 views

Bound for perimeter of convex polygon?

Is it true that the perimeter of any convex polygon in the unit disk on the Euclidean plane is less than the circumference $2\pi$ of the circle? Thanks. [The OP has solved it]
1
vote
1answer
515 views

Max Euclidean Distance between two points in a set

Given a set of Euclidean Vectors with $N$ dimensions, whose distance from a Euclidean Vector, $R$, is less than some Constant, $C$. Can the max distance between any two vectors in the set be ...
0
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0answers
24 views

Prove that CY is the external bisector of C [duplicate]

Possible Duplicate: Prove that CX and CY are perpendicular There is given convex quadrilateral ABCD. And internal bisectors of angle ∠A and ∠C intersect in point X. And internal bisectors ...
0
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0answers
127 views

Euclidean Geometry Proof using the Ruler Axiom

Link to the problem In this context, a coordinate system is given by the Ruler Axiom, which states: Let $l$ be any line. Then there is a one-to-one correspondence $f: l \rightarrow \mathbb{R}$ ...
3
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3answers
371 views

Prove that CX and CY are perpendicular

There is given convex quadrilateral ABCD. And internal bisectors of angle $\angle A$ and $\angle C$ intersect in point X. And internal bisectors of angle $\angle B$ and $\angle D$ intersect in point ...
0
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1answer
32 views

Minimial parameters to discribe a point on the surface of a high dimensional unit sphere

Consider a 2N dimensional space, $x\in \mathbb{R}^{2N}$ is a point with constraint $||x||_2=1$ Thus $x$ is actually lies on the surface of a unit sphere. Given that we know the fact $x$ is always on ...
1
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1answer
149 views

Construction of given quadrilateral

There is given convex quadrilateral ABCD. And internal bisectors of angle $<A$ and $<C$ intersect in point X. And internal bisectors of angle $<B$ and $<D$ intersect in point Y. And ...
0
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1answer
87 views

Shortest path between two points that stays away from its rotated images

In the plane equipped with an orthonormal basis, let us consider the two points $A$ and $B$ whose coordinates are $(-2,0)$ and $(1,1)$, respectively. Is there a path from $A$ to $B$ (i.e. a continuous ...
27
votes
4answers
1k views

Two squares in a box.

According to Arthur Engel, "Problem Solving Strategies", this problem goes back to Erdős, but I cannot find the solution: Let $A$ and $B$ be two non-overlapping squares inside a unit square, of side ...
1
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1answer
391 views

How to determine oriented angle of rotation in 3-dimensional space

Let $V$ be oriented two-dimensional Euclidean space. Then we can define an oriented angle $\phi$ between two nonzero vectors $u,v\in X$ by formulas: $ \phi=\arccos \frac{\langle u,v\rangle}{\|u\| ...
1
vote
1answer
264 views

Equivalence between algebraic statements and geometric relations.

I'm currently trying to read a geometry and symmetry book and came across a little problem that I am having difficulty understanding. I need to show that if xm=mx then the point X is on the line M, ...
1
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2answers
406 views

Triangle in hexagon

In a regular hexagon ABCDEF is the midpoint (G)of the sides FE and S intersection of lines AC and GB. (a) What is the relationship shared point of straight ...
0
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2answers
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 ...
1
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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 ...
1
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2answers
508 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 ...
7
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1answer
972 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
0
votes
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. ...
8
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1answer
255 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
259 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 $|BC|=l$. ...
1
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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. ...
2
votes
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 $|CA|+|CB|=2|AB|$?
0
votes
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 ...
3
votes
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
votes
1answer
460 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
608 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 ...
0
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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 ...
1
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2answers
748 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 ...
14
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6answers
790 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 $n \ge 3$)?
5
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1answer
7k 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.
3
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0answers
237 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 ...
2
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0answers
270 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 ...
2
votes
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 ?
0
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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 ...
0
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1answer
114 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 ...
0
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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, ...
1
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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?
12
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2answers
355 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 ...
3
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2answers
265 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 ...
7
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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 ...
2
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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 ...