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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Shortest path between two points via two disks

Hallo everybody, I have the following problem regarding shortest paths in $R^2$. Suppose you are given two points $p$ and $q$ and two unit disks, as in the picture. I am looking for a path from ...
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2answers
712 views

Rotations by degrees other than $90, 180,$ and $270$.

Say I have a triangle with vertices $(0,0), (2,4), (4,0)$ that I want to rotate along the origin. Rotation by multiples of $90^{\circ}$ is simple. However, I want to rotate by something a bit more ...
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1answer
47 views

exercises for Euclid's Elements

Can you suggest some books with exercises related to Euclid's Elements, or to Euclidean Geometry, as an aid to an undergraduate course on Euclidean Geometry and its history? I need exercises that ...
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4answers
1k views

Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it in a book a few years ago. Though I've tried to solve it, I'm facing difficulty. Question: You know on a plane there is an ...
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1answer
57 views

Geometry question pertaining to a plane going through the skeleton of a cube

My question is as follows: a plane that has taken the shape of a pentagon is intersecting the skeleton of the cube. Or I guess we could think of it as a cross section. Points $M$ and $N$ were used ...
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281 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
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2answers
257 views

Construction of a triangle, given: side, sum of the other sides and angle between them.

Given: $\overline{AB}$, $\overline{AC}+\overline{BC}$ and $\angle C$. Construct the triangle $\triangle ABC$ using rule and compass.
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0answers
68 views

There exist a bijection between the Real numbers and points of a straight

Assuming that we are building our geometry on the axioms of Euclid/Hilbert, and using either the Dedeking or Cauchy construction of the reals, how can one prove this statement? I've looked up on the ...
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2answers
91 views

Visually understanding the formula for the distance from a point to plane.

Ok, so we know that if we have an arbitrary point, $p$, and a normal perpendicular to an arbitrary plane, $n$, the distance from the point to the plane can be computed as follows: $$distance = p ...
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3answers
966 views

Making a convex polyhedron with two sheets of paper

Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that ...
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2answers
425 views

Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?

Let's call a point $P$ which satisfies the following condition 'a rational point'. Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ ...
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1answer
47 views

Is there a name for this point?

I found the following problem interesting: In a three villages $A$, $B$ and $C$ there are $a,b$ and $c$ pupils respectively. Where should one build the school such that the total length of pupils ...
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1answer
35 views

Distance measures for binary data

I was wondering what are some good distance measures for binary data that have the following properties. I know that there are measures like the Jaccard index and the Dice Index, but they don't ...
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0answers
5 views

Propagating a 3d vector to a spcific point in a 2d plane

I have an $xyz$ point $P$, and a 3d vector pointing from it denoted by $N$. I want to propagate the vector forward to a certain point in the $xy$ plane and calculate the corresponding value of $z$. ...
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2answers
52 views

Find the value of $a$.

please help I'm lost on what numbers to add or what formula to use
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1answer
93 views

I need to find the value of x. Im only given the a degree how would you solve this?

this is the link to the triangle that is connected to the question. What is the value x?
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2answers
554 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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1answer
251 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
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2answers
899 views

Formal Proof that area of a rectangle is $ab$

I tried to prove that the area of a rectangle is $ab$ given side lengths $a$ and $b$. The best I can do is the assume the area of a $1\times1$ square is $1$. Then not the number of $1\times1$ squares ...
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1answer
58 views

In the change-of-variables theorem, must $ϕ$ be globally injective?

In the above theorem, doesn't $\phi$ need to be injective too? The inverse function theorem merely implies that $\phi$ is locally injective -- is this sufficient? I ask because Marsden, in his ...
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2answers
51 views

How to find the area of an isosceles triangle without using trigonometry?

I have an isosceles triangle with equal sides $10$ unit, angle between them is $30^\circ$. I need to be confirmed that the area of this triangle can be found in any method other than using any kind ...
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1answer
65 views

Is there anything we can add to the present Euclidean Geometry?

I did not study the detail history of Euclid. As far as I know, the geometry starts from a few postulates and keeps on expanding (through theorems developing) to the present state. Even with these ...
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2answers
62 views

Constructing two tangents to the given circle from the point A not on it

I'm trying to complete Level 21 from euclid the game: http://euclidthegame.com/Level21/ The goal is to construct two tangents to the given circle from the point A not on it. So far I've figured ...
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0answers
13 views

Convert Euler angles from one group of rotation axes to another

I have Z-X-Z Euler angles which I would like to convert to X-Y-X Euler angles. What would be the formula for that? The exact choice of source and target rotation axis is not important, I just wish to ...
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2answers
141 views

Simple 9-th grade geometry problem

I have a geometry problem which states that Find the range of $x$ in following figure. Given that $AD$ and $AC$ are equal, and the values and angles are also given. How to estimate the range ...
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1answer
69 views

About the area of integer-edge-length triangles

Let $a,b,c$ be three edge lengths of a triangle whose area is $S$. Then, here is my question. Question : Supposing that $a,b,c$ are natural numbers, then does there exists $(a,b,c)$ such that ...
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RFC: Proof relating area to intersection of points.

this is the first time I used latex. Please excuse the rough edges. I asked a question about this problem here: Relating area to a line intersecting with a point. but I think I was able to find a ...
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3answers
164 views

Bearings Problem

I'm presented with the following bearings problem. I believe I have graphed it correctly, although I don't know where to go from here. A US Coast Guard patrol boat leaves Port Cleaveland and ...
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1answer
74 views

Distance between two barycentric coordinates

I am developing a system, and generally in this system we examine the effect of a number of factors on our data. We choose to use Barycentric coordinates to help us to achieve that. I have no problem ...
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1answer
42 views

Can any vertex of an isosceles triangle represent the centre of a circle, and the base vertices represent points on the circumference of that circle?

This question occurred to me doing this circle geometry problem, and I was wondering if anyone could clear it up. Geometrically, it seems it would make sense, provided that 2 sides are equal (equal ...
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1answer
108 views

Determine the orbits of isometries of the plane on unordered pairs of points

"Let the group M2 of isometries of the Euclidean plane act on the set S consisting of pairs of unordered points of the plane. Determine the orbits of this action, and for each orbit, the stabilizer ...
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0answers
24 views

Is there an algorithm that, given a point cloud, infers an optimal wireframe (surface) structure?

I have a point cloud that I would like to convert to a surface, in the form of a wireframe lattice structure. This means, from a sequence of 3D points (x,y,z), obtaining three 2D matrices X,Y,Z of ...
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4answers
455 views

If any triangle has area at most 1 , points can be covered by a rectangle of area 2.

I am working on this problem for some time, and I am not able to finish the argument: There is a finite number of points in the plane, such that every triangle has area at most 1. Prove that the ...
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1answer
41 views

dot product of vectors with not orthogonal basis

The dot produt (inner product in the context of Euclidean space) of two vectors $\mathbf{a}=\left [ a_{1},a_{2},...,a_{n} \right]$ and $\mathbf{b}=\left [ b_{1},b_{2},...,b_{n} \right ]$ is defined ...
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1answer
39 views

Relating area to a line intersecting with a point.

I really could use a hint with this following problem: If a line L separates a parallelogram into two regions of equal areas, then L contains the point of intersection of the diagonals of the ...
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2answers
56 views

what is the area of face of the cube in $m^2$

A fly is trapped inside a hollow cube. It moves from A to C along edges of cube, taking shortest possible route. It then comes back to A again along edges, taking longest route(without going over any ...
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1answer
143 views

Using Square area finding quadrilateral area

Area of square ABCD is 169 and that of square EFGD is 49. Find area of quadrilateral FBCG I am stuck and just thinking which way can be helpful for me finding this area of quadrilateral FBCG. ...
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2answers
216 views

Finding the third point of an equilateral triangle in three dimensions. [closed]

The coordinate of A is (4,-3,5) and the coordinate B is (6,7,8). Find the coordinate of C such that triangle ABC forms equilateral triangle It is easy to work in two dimension Cartesian ...
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1answer
81 views

Incentre and excentre of a triangle

Prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to these sides is equivalent to the triangle formed by the points of contact ...
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0answers
55 views

Equivalent descriptions of “flat space with non Euclidean metric” and “curved space with (local) Euclidean metric”: the case of Minkowski space.

FIRST: I start with the guiding idea: 1. we have the surface of a paraboloid (z = x2 + y2); its metric, in an infinitesimal neighbourhood of one of its points is (we can choose it) EUCLIDEAN; now, ...
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3answers
74 views

Equation of rectangle

I need equation of a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? for example equation of ellipse is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
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1answer
36 views

Function on plane with incenter

Let $f$ be a function from the set of all points on the plane to the nonzero real numbers. Suppose that for any triangle $ABC$ with incenter $I$, we have that $f(I)=f(A)f(B)f(C)$. What are the ...
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1answer
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Solving the following relation in triangle

If a line through the centroid $G$ of a triangle ABC meets $AB$ in $M$ and $AC$ on $N$ then prove that $AN. MB+AM. NC=AM. AN$ both in magnitude as well as sign I tired dividing the equation by $AM. ...
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0answers
24 views

Bisector of a triangle

From A, perpendiculars AX, AY are drawn to the bisectors if the exterior angles of B and C of triangle ABC. Prove that XY parallel to BC
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1answer
53 views

Parallelogram constructed through medians

Bdmo In $\Delta ABC$, Medians AD and CF intersect at P.Let Q be any point on AC.Construct QM and QN parallel to AD and CF respectively.Now the line joining M and N intersects CF and T and AD at ...
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1answer
41 views

Doubts about locus and its equation

Two points A and B with $(1,1)$ and $(-2,3)$ respectively are given.find the locus of point P.So that area of $\Delta$PAB is 9 square units. answer is :- $2x+3y+13=0$ or $2x+3y-23=0$. how i tried:- i ...
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3answers
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How can I find the volume of prism: $V = \frac{(a + b + c)Q}{3} $

In the book Handbook of Mathematics (I. N. Bronshtein, pg 194), we have without proof. If the bases of a triangular prism are not parallel (see figure) to each other we can calculate its volume by ...
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2answers
75 views

Geometry : find the points of tangency between two lines and two circles [closed]

I have a programming problem. I need to find the intersection points between two lines tangent to two circles and the circles! I have the circles' radiuses and centers. So I need points ...
3
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3answers
82 views

Average distance from a point in a ball to a point on its boundary

What variety of methods are readily available to find the average distance from a point in $\{ (x,y,z) : x^2+y^2+z^2 \le r^2 \}$ to the point $(0,0,r)$? I just worked this out and got $6r/5$. Later ...
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1answer
51 views

How to embed this circle tangent to the other circles?

I want to construct a circle that would be tangent to the $3$ circles and would have its diameter lie somewhere on the segment $BI$. $EF$ includes the diameters of the $3$ given circles. $EB=BF$. ...