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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35 views

The Least Area For a Needle to Pass Through a Curve?

I don't know if this question is a famous one. One of my fellows asked me these questions to tease me, but I was able to find a solution for only one of these: There is one needle of length $2$ ...
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18 views

Points collinear proof

Diagonals of quadrilateral $ABCD$ intersect in point $S$. Circle $k_1$ is circumscribed of triangle $ABS$. $k_1$ intersects line $BC$ in point $M$. Circle $k_2$ is circumscribed of triangle $ADS$ and ...
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4answers
32 views

prependicular Vs prependicular bisector

We have $AH=HB$ and $BG=GC$ in the image below. Why is $AD=2\times FG$?
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36 views

Staircase Lemma

Let $S$ be a staircase-shape contained in the north-eastern quarter-plane. Let $k$ be the number of its south-western corners. In the staircase shown below, there are $k=4$ corners: In each corner ...
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2answers
28 views

Is it possible for Euclid's theorem to generate 3 4 5?

I am writing an algorithm to generate pythagorean triplets and I was going to use Euclid's theorem, however I have been unable to make it generate the first pythagorean triplet namely (3,4,5). Is this ...
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1answer
17 views

How to compute an angle in specific counter-clockwise direction between vectors

I have one incoming vector and multiple outgoing vectors in 2D. I need compute an angle in this way: Imagine an incoming vector parallel to the x-axis. Then the angle-value "starts" below the ...
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0answers
25 views

To construct a right triangle given the hypotenuse and sum of two legs [duplicate]

NOTE: I want a hint only. A compass and a straightedge construction:Given a hypotenuse and the sum of lengths of the legs,we need to construct a right triangle. MY TRY: From any ray $BE$, ,let ...
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0answers
11 views

Finding Euler path/circuit of detached items?

I'm trying to find the Euler path of the item below, however, I'm not sure how to do so because the middle item is detached. Does anyone know what to do here?
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1answer
32 views

Draw a picture of a cube with edges a+b, and show it cut by planes that divide each edge into a segment of length a and a segment of length b.

I am reading through 4 pillars of geometry and I need some help with this question. Draw a picture of a cube with edges a+b, and show it cut by planes (parallel to its faces) that divide each edge ...
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4 views

Updating vector components based on change of starting position

I have an xyz point, and a 3D vector originating at that point. I would like to be able to shift the starting xyz point and update the 3D vector accordingly. For example: Starting at the xyz point ...
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2answers
36 views

Maximize the inradius given the base and the area of the triangle

BdMO 2013 Secondary: A triangle has base of length 8 and area 12. What is the radius of the largest circle that can be inscribed in this triangle? Let $A,r,s$ denote the area,inradius and ...
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3answers
54 views

Equation of a line passing through a given point, perpendicular with a line [closed]

I need the equation of the line passing through a given point $A(2,3,1)$ perpendicular to the given line $$ \frac{x+1}{2}= \frac{y}{-1} = \frac{z-2}{3}. $$ I think there must bee some kind of rule ...
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3answers
70 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
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1answer
30 views

Heptagonal tesselations

Are there any tesselations of the Euclidean plane that use only regular polygons such that one of them is a heptagon? If so, what is the tesselation that uses the fewest different types of polygon ...
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1answer
10 views

Calculating incremental coordinate change along a 3D vector

This is pretty trivial, but I've not managed to resolve it to my satisfaction and it has reached the stage where fresh eyes are definitely useful! I have an xyz point, and a 3D vector originating at ...
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1answer
62 views

What class am I most prepared for?

I've only taken up to calc 3, discrete, and linear algebra. Which course am I most prepared for? I'm going to be taking differential equations and advanced calc, but I want to take a 3rd class. I can ...
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4answers
689 views

which axiom(s) are behind the Pythagorean Theorem

There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the ...
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2answers
39 views

Prove $\sin^2 A = \sin^2 B \sin^2 C - 2\sin B \sin C \cos A$

I am asking for help with this proof: Given $\triangle ABC$. Prove that $\sin^2 A = \sin^2 B+ \sin^2 C - 2\sin B \sin C \cos A$
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2answers
47 views

Is there a Taylor series for vector cross product?

I have this equation, where $u,v,w,a,b,Ɵ$ are constants. The RHS comes from the Geometric definition of the LHS $(u,v,w)(a,b,c)=||(u,v,w)||||(a,b,c)||\cos(\theta)$ Expanding the 2-norms ...
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4answers
108 views

Maximizing the Magnitude of the Resultant Vector

Given a set of $n$ two-dimensional unit vectors: $\left\{ \mathbf{v}_1, \dots, \mathbf{v}_n \right\}$, I want to find the coefficients $\left\{ \alpha_1, \dots, \alpha_n \right\}$, $0 \leq \alpha_i ...
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0answers
29 views

Geometric conditions equivalent to a set being the unit circle for some norm

Here's the question, as in the textbook (Real Mathematical Analysis, Pugh). The unit ball with respect to a norm $||\, \cdot \,||$ on $\mathbb{R}^2$ is $$ \{ v \in \mathbb{R}^2 : ||\, v \,|| ...
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2answers
84 views

Solve the following problem…

My problem is: In a circle of radius $R$ is inscribed an equilateral triangle $ABC$. Through the point $C$ is drawn a line which intersects $AB$ in point $M$ and the circle, for the second time, in ...
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0answers
25 views

How to prove that P, F, and E are collinear from this following parallelogram problem?

Inside parallelogram $ABCD$ with $\angle A=90^\circ$, a circle with diameter $AC$ intersects $CB$ and $CD$ at $E$ and $F$ respectively. Tangent line of this circle at $A$ intersects $BD$ at point ...
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1answer
29 views

When two triangles have the same orthocenter and circumscribing circle, are nine points are the same too?

When two triangles have the same orthocenter and circumscribing circle, are the nine points are the same too? If two triangles have the same circumscribing circle, at least the sides have the same ...
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2answers
27 views

Triangles with common centroid

Consider the points $A',B',C'$ on the sides $BC,CA,AB$ of a triangle $ABC$ respectively, such that $BA'/A'C=CB'/B'A=AC'/C'B$. Show that the triangles $ABC$ and $A'B'C'$ share a common centroid.
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2answers
39 views

how to find three vertices of a triangle.

Where s=circumcenter, H= orthocenter, and A'= midpoint of one side of triangle. How can can I determine the location of the three vertices of the triangle?
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2answers
186 views

About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
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0answers
33 views

Solution for the value of an angle of a triangle ABC

Find value of angle m< DBC Where $$BD=DC=AC$$ $$2(m\langle BAC)=14(m\langle ABD)=7(m\langle BCD)$$ I tried hard but im out of ideas now, I know the answer is 20 but I want to know how, thanks ...
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1answer
18 views

Perimeter of the triangle

$BO$ and $CO$ are angle bisectors of triangle $ABC$. How much is perimeter of the triangle $AMN$?
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21 views

Partition of the 3d space with circles?

Does it exist a partition of the 3d space with circles of positive radium? I know the answer is no for a plane, but I can not transpose my demonstration to the space and I have no clue on how to do ...
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118 views

Find features in a Signed Distance Field

I'm currently trying to improve a meshing algorithm for signed distance fields (which are of the form $f(x,y,z) = w$, where $x,y,z$ is the location of my query, $w$ indicates the distance to the ...
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1answer
50 views

How long is the diagonal of this trapezoid?

Given a trapezoid $abcd$, with $|ab| = 1$, and angles $\angle dab = 3\theta/4$, $\angle abc = (\pi + \theta)/2$, $\angle bcd = (\pi - \theta)/2$, and $\angle cda = \pi - 3\theta/4$ (see figure below), ...
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1answer
67 views

Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds

I am confused by some definitions. Forgive the looseness of my language. A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into ...
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1answer
44 views

Proving the inequality of Cauchy-Schwarz in an Euclidean space. [duplicate]

It says let (G, <.,.>) be an euclidean space. Show that for all x, y belonging to G: modulus<x,y> <= sqrt<x,x> * sqrt<y,y> and in the ...
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0answers
21 views

How to prove that the definition of saddle surface is affine invariant?

I have a smooth saddle surface in $\mathbb{R}^n$, so for any normal vector the second fundamental form of the surface has $\det \leq 0$. How can I proove that the surface is still saddle if I stretch ...
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2answers
48 views

Is a quadrilateral with one pair of opposite angles congruent and the other pair noncongruent necessarily a kite?

If convex quadrilateral ABCD has congruent angles A and C and the other pair of angles B and D are not congruent, is ABCD necessarily a kite (two pairs of consecutive congruent sides but opposite ...
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1answer
36 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
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0answers
16 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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1answer
50 views

Sylvester-Gallai Theorem

How is this theorem used in applications? I've been searching for it on the web but can't seem to find. Only to "correct codes". Can someone please give a few simple examples? /lost student
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2answers
28 views

Connecting square vertexes with minimal road

I have four cities in $A=(0,0),B=(1,0),C=(1,1),D=(0,1)$. I am asked to build the shortest motorway to connect the cities. How can I do that? I was thinking that first I need some compactness argument ...
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2answers
22 views

Orientation of vector relative to other vector

Given two directional vectors in 2D space, $\vec v=(v_x, v_y)$ and $\vec w=(w_x, w_y)$, what is the easiest way to calculate if $\vec w$ is orientated clockwise or counterclockwise relative to $\vec ...
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1answer
67 views

Are Euclidean distances a monotone function of inner products?

Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of $k$ vectors happens to decrease? Similarly-if decrease is ...
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1answer
38 views

Tangents to a circle

For this construction, how would you show that the perimeter of the triangle $CDF$ is equal to $2BC$? Please include steps and whatnot.
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1answer
40 views

volume of the solid

Using geometry, calculate the volume of the solid under $z = \sqrt{49- x^2- y^2}$ and over the circular disk $x^2+ y^2\leq49$. I am really confused for finding the limits of integration. Any help?
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1answer
36 views

Nature of Points and Lines in Euclidean Geometry

It may be true that very few middle school student can grasp the meaning of lines and points in Euclidean geometry prior to a direct instruction. For example, it's possible that such a conversation ...
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1answer
95 views

A problem with concyclic points on $\mathbb{R}^2$

I am thinking about the following problem: If a collection $\{P_1,P_2,\ldots,P_n\}$ of $n$ points are given on the $\mathbb{R^2}$ plane, has the property that for every $3$ points $P_i,P_j,P_k$ in ...
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0answers
26 views

Are a half-disk and a quarter-disk scissors-congruent?

Call two measurable subsets of the plane of equal area scissors-congruent if one can be decomposed into a finite number of measurable pieces which may be reassembled to yield the second. Any polygon ...
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1answer
249 views

Expected value of the distance between 2 uniformly distributed points on circle

I have the following problem (related to Bertrand): Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference. Then connect these points using a line with length $b$. ...
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1answer
52 views

To draw a straight line tangent to two given ellipses

How can I draw a a straight line that touches two ellipses? There are, like for two circles, 4 different solutions. I´m not interested in the analytical solution, but in the geometrical drawing, ...
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0answers
31 views

Conic equation from cone/plane intersection

In an orthonormal cartesian frame $(O; \vec{x}, \vec{y}, \vec{z})$ consider: an infinite plane $P$ defined by: a point $p = (p_x, p_y, pz)$ an normal vector $\vec{n} = (n_x, n_y, n_z)$ a cone $C$ ...