Tagged Questions

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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0answers
8 views

Does this notion of the “directed area” of a closed curve in $\mathbb R^3$ have a standard name?

Given an oriented surface $\Omega$ in $\mathbb R^3$, consider the quantity $\mathbf A=\int_\Omega\hat n\,\mathrm dA$. We may call this the "directed area" of the surface because, when $\Omega$ is ...
-1
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0answers
8 views

mobius tranformation on the extended complex plane

write the result of translating by 4+i and then carrying out the inversion as a single mobius transformation on the extended complex plane? Someone help please...
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0answers
16 views

fluid flow through an orfice [migrated]

Forgive me for my ignorance. What would be the method to determine the pressure a non compressible fluid creates when forced though an orifice? Keep in mind this orifice does not have a constant ...
0
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1answer
23 views

Area of a triangle inside a larger triangle

It's been a while since I've done any geometry so I'm a bit confused by this question. We have a triangle $\triangle PQR$ whose total area is $90 \mathrm{cm}^2$. Another triangle $\triangle PTU$ is ...
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1answer
33 views

geometry problem [on hold]

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
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1answer
22 views

Find the equation of the ellipse with given foci and $a$

I have to find the equation of the elipse with foci: $$(-1,-1),(1,1)$$ and $a = 3$ I could do it using the definition of elipse, which I know how to work with. But I need to do it using translation ...
2
votes
2answers
28 views

Sum of squares of side lengths of a regular polygon [on hold]

Given $n$ vertices, $P_i$, $i \in \{1\dots n\}$ of a regular polygon on the unit circle (radius $R = 1$), calculate the sum $|P_1P_2|^2 + \dots + |P_1P_n|^2$.
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1answer
18 views

geometry proof - Generalities of Geometry

Can you give a hint on how to prove that AB + CD > AF + FC +DE + EB?
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0answers
7 views

path metrics without geodesics

This is a follow-up of this question. Recall that a metric space $(X,d)$ is called a path-metric space if the distance between any two points in $X$ equals the infimum of lengths of paths between ...
8
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1answer
79 views

A geometric assembly: Triangle, circle, square, pentagon.

Let say we have an equilateral triangle and I draw its circumscribed circle, to continue we draw a square in which the previous circle is inscribed. After that we draw the circle circumscribed to the ...
1
vote
1answer
32 views

Do closed submanifolds have nonzero curvature on a non-zero-measure set

If $M\subset \mathbb{R}^n$ is a $m$-dimensional submanifold, does the set with non-zero gaussian curvature always have $m$-dimensional Hausdorff measure greater than zero? For a manifold homeomorphic ...
3
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0answers
17 views

Bounded area for any triangle formed by polygons

Let $P_1,P_2,P_3$ be closed polygons on the plane. Suppose that for any points $A\in P_1$ (meaning $A$ can be inside or on the boudary of $P_1$), $B\in P_2,C\in P_3$, we have $[ABC]\leq 1$. Is it ...
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1answer
15 views

Finding Orthocenter in Coordinate Geometry

If a triangle is formed by the equations \begin{gather}2x+3y-1=0\\ ~~x+2y-1=0\\ ax+by-1=0\end{gather} and has its orthocentre at origin, then what are the values of $a$ and $b$? (Please also tell me ...
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0answers
33 views

What is the average distance from the base of a rectangular pyramid to its apex?

What is the formula for the average distance from the base of a rectangular pyramid to its apex? For example, if the base of the pyramid is 30 feet by 8 feet, and the height of the pyramid is 12 feet, ...
3
votes
0answers
14 views

Product to vertices in triangle maximal

Suppose we're given a triangle $ABC$. At which interior point $T$ is the product of distances $|AT|\cdot |BT|\cdot |CT|$ maximal? Is it a known point, like the centroid or incenter?
0
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0answers
14 views

Abother question on game theory [duplicate]

There is a polygon with $N$ vertices drawn in the plane. The polygon is strictly convex, i.e., each internal angle is strictly smaller than 180 degrees. The vertices of the polygon are numbered 1 ...
0
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1answer
19 views

Understanding why Euler's Formula applies to planar graphs

I'm trying to prove that given a planar graph (by that I mean a graph where every pair of points is joined without crossings) $V-E+F = 2$. I can prove this by induction directly on the edges except ...
4
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1answer
58 views

Selecting a Random Point Inside a Cube

A point $P$ is selected at random inside a cube. Find the probability that $\angle APB \geq 135^o$, where $\overline{AB}$ is a body diagonal of the cube. I am not able to come up with the right ...
2
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2answers
56 views

Closed-form of $\tan(\frac{\pi }{9})$.

How can I get the closed-form of the above expression.I tried to find it but I can't do.
1
vote
1answer
37 views

Finding circle with two points on it and a tangent from one of the points

Two points P1(x1,y1) and P2(x2,y2) are known. In addition, a line slope passing through P1 is known. The aim is to construct a circle (or circular arc) that it passes through both P1 and P2 and it is ...
0
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2answers
16 views

What is the angle of a space diagonal in a cuboid?

What is the angle ($\sigma$) from $D$ to $R$, expressed as function of $R$, $T$ and $H$?
0
votes
1answer
17 views

Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?

I'm trying to construct a homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$. I'm pretty sure there is one. I've been trying to work geometrically : mapping $[0,1]\times[0,1]$ to ...
2
votes
2answers
53 views

Methods for calculating $\pi$ that use the sphere?

The area of the unit circle is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as ...
1
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1answer
14 views

A clarification on a problem on proving existence of regular polygons

So I was looking at this thread: How can I prove the existence of an octagon/decagon/dodecagon?, and while this question intrigues me I also have no way to solve it, and it was unanswered in the ...
0
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2answers
27 views

How to calculate the curvature of a curve whose equation is not given.

I want to calculate the curvature of the following curve (in blue) whose equation is not known. I shall be thankful.
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3answers
35 views

How to show whether three points in $\mathbb{R^4}$ lie on a straight line?

If you are given the coordinates of three points in $\mathbb{R^4}$. (Call these three points: A, B and C). How do we know whether these three points lie on a straight line or not? One way I can think ...
0
votes
1answer
19 views

Spherical Sector Volume

I'm trying to find the volume of a spherical sector without knowing the height of the cap. Wikipedia provides this formula: And says: "where φ is half the cone angle, i.e., the angle between the ...
0
votes
1answer
21 views

Problem with shaded regions in a square

The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles (diagram 1). The process is repeated twice as shown in the ...
0
votes
1answer
28 views

Prove: sum of angles in triangle is invariant

Prove that the sum of angles in any plane triangle is invariant, without invoking the fact that the sum of angles in a triangle is two right angles. I've tried. Is it even possible?
3
votes
3answers
56 views

Why are randomly drawn vectors nearly perpendicular in high dimensions

I am struggling understanding this finding. Can somebody explain intuitively why randomly drawn high-dimensional vectors will tend to be mutually orthogonal? I realize that intuition in high ...
0
votes
1answer
18 views

Prove that a convex $d$-polytope has at least $d+1$ facets

This seems trivial but I can't come up with a formal proof. I think there should be a way to do this inductively but I can't figure out how$\ldots$ Any help much appreciated
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2answers
37 views

Maximal points on an $n$-dimensional ellipsoid

For an $n$-dimensional ellipsoid in $\mathbb{R}^n$ centered at $\mathbf{0}_n$, defined by a set of vectors $\mathbf{v}_i$ giving the directions of the principle axes and $\lambda_i$ representing the ...
4
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1answer
35 views

prove this properties of triangles trigonometric question

The triangle $DEF$ circumscribes the three escribed circles of triangle $ABC$. Prove that $$\frac{EF}{a\,\cos A} = \frac{FD}{b\,\cos B} = \frac{DE}{c\,\cos C}$$
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0answers
21 views

Interior of sum of sets equals sum of interior of summands

I'd like to have the answer to the following question. If $X_1,X_2\subseteq \mathbb{R}^n$ are convex and compact sets of dimension $n$, does the following hold: ...
2
votes
3answers
33 views

Drawing a Cone on a Plane

If we draw a cone on a plane, the picture usually consists of an ellipse, two line segments sharing one common endpoint, while the other end point connected to each vertex of major semi-axes of the ...
0
votes
1answer
11 views

How to find the reflection matrix

$V$ is an m dimensional subspace of $\mathbb{C}^n$ , n>m with an orthonormal basis {$q_1$,..,$q_m$}. How to find the reflector $P\in \mathbb{C}^{nxn}$ that reflects about $V$. $P$ must depend on ...
0
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1answer
32 views

finding the area of a trapezoid using only 2 of the 4 triangles that makes up its interior

Given a trapezoid $ABCD$ with its diagonals drawn and $E$ is the point where the diagonals intersect. Then the trapezoid is divided up into 4 triangles. Theres a well known theorem that if ...
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0answers
35 views

Find the radius of a circle [on hold]

An equilateral triangle has an inscribed circle which contains another equilateral triangle inscribed in it. The difference between the areas of the triangles is 40 cm2. Find the radius of the circle. ...
3
votes
1answer
25 views

Triple integral over an ellipsoid

Let $E$ be the solid ellipsoid $E = ${$(x,y,z)$ | $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \le 1$} where $a > 0,\: b > 0,\: c > 0$ Evaluate $\int\int \int xyz\: dxdydz$ over: a. ...
0
votes
1answer
18 views

Intuitive understanding of relationship between unit vectors and position vector

On Wolfram Mathworld they give a unit vector in the $\textbf{x}_n $ direction as: $$ \hat{\textbf{x}_n} \equiv \dfrac{\frac{\partial \textbf{r}}{\partial x_n}}{\lvert \frac{\partial ...
6
votes
1answer
43 views

Intuition behind a certain limit.

We want to find $\displaystyle\lim_{\theta\to\frac{\pi}{2}} b_1-a_1$, we are given $c=1$ and that $\cdot=90^{\circ}$ This is my solution; $$\begin{equation}\sin \theta=\frac{b_1}{a_1} \iff b_1=a_1 ...
10
votes
0answers
62 views

How to prove that the problem cannot be solved by the four Arithmetic Operations?

The original prolbem is as in the figure: Suppose the square has unit side length, find the area of blue region. The exact solution is: $$\begin{aligned}S=&\frac{\pi-\sqrt{7}}{4}+2 ...
0
votes
0answers
24 views

Finding a way between two points

Let's assume we have two points $A(x_1,y_1)$ and $B(x_2,y_2)$ in 2-D space. And we need to find a trajectory for going from one point to the other. But the problem is that in this space prohibiting ...
0
votes
2answers
16 views

Convert a triple integral to cylindrical coordinates?

Find the volume determined by $$z \le 6-x^2-y^2$$ and $$z \ge \sqrt{x^2+y^2}$$ I used cylindrical coordinates to change the bound for $z$ to $r \le z \le 6-r^2$. However, I am not sure how to find ...
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vote
2answers
22 views

How to compute volume of a circle defined by L1 distance?

In n dimension space, given a central $x=(x_1,x_2......x_n)$ and radius r, a circle C is defined as all point $y=(y_1,y_2,.....y_n)$ satisfy $ \sum_{i=1}^n\left\lvert y_i-x_i\right\rvert <= r$ ...
0
votes
2answers
16 views

Two correspoding sides of similar triangles are perpendicular

I don't understand why if two pairs of corresponding sides of similar triangles are perpendicular, the third pair of corresponding sides must be perpendicular as well. Could someone prove it/ explain ...
0
votes
2answers
25 views

Finding the bounds on a triple integral

Problem: Find the volume enclosed by the cone $$x^2 + y^2 = z^2$$ and the plane $$2z - y -2 = 0$$ So I know that I need to do a triple integral over this region, and the integrand will be 1. My ...
1
vote
1answer
43 views

A triangle ABC with vertex $C(4,3)$. The bisector and the median line equation drawn from the same vertex are given. Find the vertices A & B.

A triangle $\triangle ABC$ one of his vertex is the point $C(4,3)$. The bisector line equation is $x+2y-5=0$ and the median line equation is $4x+3y-10=0$ drawn from the same vertex. Find the ...
0
votes
1answer
6 views

Determining the distance across a rectangle for an arbitrary angle

I'm trying to determine the 'diameter' or distance across a given rectangle for a given angle. Ideally there should be a function f(theta) that takes a polar angle and determines the distance across ...
0
votes
0answers
33 views
+50

Rank reduction to satisfy Barvinok's upper bound & Rank of a set notation

After reading n times the four first sections of the 4th Chapter of J.Dattorro's book (Convex Optimization & Euclidean Distance Geometry). I am confused between yes or no, every extreme point of ...