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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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 ...
13
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3answers
938 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
395 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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0answers
58 views

Euclidean Geometry [on hold]

Let $ABC$ and $A'B'C'$ be two non-congruent triangles whose sides are respectively parallel . Then prove that $AA',\, BB', \, CC'$ (extended) are concurrent. Look I came across this problem in a ...
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0answers
42 views

Euclidean Geometry of a triangle [on hold]

Let p and q be radii of two circles through A, touching BC at B and C respectively. Then prove that pq = $R^2$ . Actually I got this in book and there also it was not clearly mentioned about about R ...
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1answer
45 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
32 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
44 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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0answers
57 views

Examples of Beautiful Applications of Thale's Theorem [closed]

My question is this: what do you consider to be a particularly beautiful application of Thales' Theorem? I'd like to collect a bunch of nice examples as a big list. One of my favourites, as a ...
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1answer
45 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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1answer
246 views

Bounding box enclosing circles, that complies with ratio constraints

Given a circle centered at $A$, with radius $R_a$ and another radius $R_b$, I need to find a center for circle $B$ such that both circles are tangential, and the bounding box including both circles ...
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2answers
485 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
205 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
768 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
47 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 ...
2
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2answers
36 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 ...
3
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1answer
574 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 ...
2
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1answer
582 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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1answer
54 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
157 views

Expressing an isometry [closed]

Let $s$ denote reflection of the plane about the vertical axis $x=1$. Also let $r$ dentoe the reflection with respect to the horizontal component of the basis in $\mathbb R^2$. Find an isometry ...
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2answers
37 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
10 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 ...
2
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2answers
126 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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0answers
26 views

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
66 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 ...
3
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1answer
33 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
27 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
103 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
19 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
439 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
22 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
36 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
54 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
128 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
126 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
27 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
34 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
69 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
26 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
14 views

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
16 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
3
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0answers
53 views

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2. What is the smallest possible angle of the polygon? What is its smallest possible area? ...
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1answer
288 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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1answer
46 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
38 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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1answer
787 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!
3
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3answers
96 views

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
46 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 ...