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

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4
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0answers
33 views

A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
1
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1answer
25 views

Projection of the XY plane.

Is there a way to project the infinite Complex plane to either the Poincare disk or the unit disk - for all values of x + iy ?
0
votes
1answer
32 views

A formula to calculate the partial volume of a capsule or tank?

We are trying to ascertain the correct formula discussed in this post. The volume formula for a capsule (a cylinder with a hemisphere at both ends) is, $$V_c = \pi r^2 H + \frac{4}{3}\pi r^3\tag1$$ ...
0
votes
1answer
30 views

Find the value of $x$ below

$AB=DC$, Find the value of $x$ I tried with Law of Sines but I get different answer every time
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0answers
26 views

$D, E, F$ are respectively projection of $O$ on $BC, CA, AB$. Prove that $\cot{\angle ADB} + \cot{\angle BEC} + \cot{\angle CFA} =0$

Let $O$ is located in the triangle $ABC$. $D, E, F$ are respectively projection of $O$ on $BC, CA, AB$. Prove that $$\cot{\angle ADB} + \cot{\angle BEC} + \cot{\angle CFA} =0$$
-5
votes
0answers
26 views

Geometry problem… [on hold]

GH=OP Find $x$ [1 any help guys ??
0
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0answers
21 views

Generalization of Lines and Planes

Let $a_1,a_2,\dots,a_n$ be constants that are not all zero. An equation defines a line if and only if it can be written: $a_1x_1+a_2x_2+a_3=0$ An equation defines a plane if and only if it can be ...
0
votes
2answers
16 views

If the area of a trapezoid is 63 square feet, find the height. [on hold]

The bases of a trapezoid are one and three feet longer than the height respectively. If the area is 63 square feet, find the height.
1
vote
2answers
29 views

$\frac{y-b}{r}=\frac{y}{s}$ to $y$ for finding the closest point on a line, from a point.

$$r=sy^2-sby$$ How do I get $y$ on one side? Originally I had: $\dfrac{y-b}{r}=\dfrac{y}{s}$
1
vote
1answer
32 views

Arbitary plane curve.

Does an arbitrary curve in the plane necessarily pass through a rational point? That is, a point of the form $(a,b)$ where $a$ and $b$ are rational numbers.
3
votes
2answers
19 views

Does the radius of the quadrant pass from the center of the inscribed circle?

In the following picture: The smaller circle is inscribed inside the quadrant, whose radius (OB) is 8. The original question (but not the question of this post) is that "find the radius of the ...
1
vote
0answers
35 views

Create solid torus with geometric algebra

I have created an algorithm for tracking a vessel's centerline in 3 dimensions, and traveling through that centerline. My supervisor asked me whether I could add a small theoretical section on ...
1
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2answers
18 views

Create Ellipse From Eccentricity And Semi-Minor Axis

So I am given the eccentricity of an ellipse and the radius semi-minor axis as well as the center of the ellipse. So in the example below we know the center of the ellipse is at ( 0, 0 ) and the ...
-1
votes
0answers
18 views

How to find truncated cylinder\ungula Volume

How would I calculate the volume at a height in an upside down truncated cylinder? Everything I find online shows a truncated cylinder being flat and level on its base, but what if the cylinder is ...
2
votes
2answers
44 views

O.I.M. polygon inequality

I am trying to prove an inequality which was used to prepare the Romanian O.I.M. team. I seem to lack ideas on how to tackle this problem. We take a convex polygon $P_1\ldots P_{n+2}$ and consider ...
1
vote
0answers
37 views

How do I calculate the area the Wigner Seitz cells cover in a square?

It's my first time here, so I appologise in advance if I break any rules through this post. So I have a Cartesian Lattice spanning across the Euclidean plane and a unit square. The lattice points ...
2
votes
1answer
43 views

One special case of Helly's theorem (for $\text{radius}=1$ circles)

There are $n$ points on the plane. Any $3$ of them can be covered with a radius $1$ circle. Prove that there is a radius $1$ circle that covers all the points. Came to this when tried to prove an easy ...
1
vote
4answers
80 views

Find the area of a triangle given the radius of its incircle and a tangential point

A friend gave me recently the following interesting problem and I would like to share a couple of solutions. Any additional contributions are welcome. A triangle $\vartriangle ABC$ is given and we ...
1
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0answers
27 views

Intuitive way to find the minimum surface in $\mathbb R^3$?

Suppose we have two arbitrary closed curves which intersect neither each other nor themselves. By intuition, I guess that the minimum surface ending at boundaries is unique and it is achieved by this ...
1
vote
1answer
13 views

Countable intersection of Cut Locuses is always empty?

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then does there always exist a countable collection of points $\{p_n\}_{m \in \mathbb{N}}$ such that: \begin{equation} ...
0
votes
2answers
30 views

Width of rotated plane

I'm trying to get the width of a rotated plane, but my knowledge of trig functions didn't really help me get what I want. I have a plane, that is $310$ units wide, and is $200$ units away from the ...
0
votes
1answer
27 views

inner product of positive semi definite symmetric matrices [on hold]

I have a positive semi definite symmetric matrix $X$, $(n\times n)$. let $X=vv^T$ s.t $\|v\|=1$. I came to a point where I am stuck to show which is: $v^TYv=\langle X,Y\rangle$ (How to show this ...
1
vote
3answers
50 views

Regular n sided polygon

$A_1A_2A_3....A_{18}$ is a regular 18 sided polygon.B is an external point such that $A_1A_2B$ is an equilateral triangle.If $A_{18}A_1$ and $A_1B$ are adjacent sides of a regular n sided polygon.Then ...
2
votes
1answer
35 views

Circle bisecting the circumference of another circle

If the circle $x^2+y^2+4x+22y+l=0$ bisects the circumference of the circle $x^2+y^2-2x+8y-m=0$,then $l+m$ is equal to (A)$\ 60$ (B)$\ 50$ (C)$\ 46$ (D)$\ 40$ I don't know the condition when one ...
0
votes
1answer
30 views

What condition do I have set to have $x_j$ for $j=1,…,n$ to be non-negative?

I have $\sum_{j=1}^n q_{jj}(x_j-y_j)^2\le1$ What condition do I have set to have $x_j$ for $j=1,...,n$ to be non-negative? The book I am reading says $\sqrt{q_{jj}}\ge1/y_j$ but why? edit: ...
3
votes
1answer
35 views

Prove a rotating window shade won't break my window when raised to a specific height.

I have a large lampshade that covers my window to block out sunlight. It has a metal rod sewn in at the bottom to weigh it down, but it's aluminum, so it can rock in wind. We recently had a flash ...
0
votes
0answers
38 views

Segments of a hypotenuse

The hypotenuse of a right triangle is divided into 2 segments by the altitude to the hypotenuse The sum of the greater segments on the hypotenuse of 2 disimilar right triangles is equal to the ...
0
votes
2answers
40 views

Can you do this to find circumference from area of a circle

If you divide the circumference by $2$, does it equal the area divided by the radius? That is, do you have $C/2 = A/r$ for any circle? ...
1
vote
2answers
15 views

Geometrical calculation to enlarge the height of rotated rectangle

There is a polygon (rotated rectangle) that defined by 4 corner points in 2D coordinate system. Does anyone help me with the fast (minimum trigonometry operations) algorithm to change its height by ...
1
vote
1answer
27 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
1
vote
1answer
38 views

'Chasing sides' in a geometry problem

Consider the circle $W=x^2+y^2=81$. Let $AB$ be a diameter of circle $W$. $AB$ is extended through $A$ to $C$. Point $T$ lies on $W$ so that line $CT$ is tangent to $W$. Point $P$ is the foot of the ...
3
votes
1answer
27 views

Intersection of Random Subspace and Hypercube

Suppose that $A \subset \mathbb{R}^n$ is a random linear subspace of dimension $k < n$. I am interested in the event that $A$ intersects the hypercube $[-1,\ 1]^n$ at a specific face. In other ...
0
votes
0answers
39 views

Triangle inscribed in another triangle

If $a,b,c$ are the sides of a triangle,$\lambda a,\lambda b,\lambda c$ the sides of a similar triangle inscribed in the former and $\theta$ the angle between the sides $a$ and $\lambda c$,prove that ...
-1
votes
0answers
9 views

2 dimension riemann manifolds of signature 0 metric

Does anyone have a proof that any 2d riemann manifold is conformally flat if metric has signature 0? Thanks.
0
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1answer
58 views

Evaluate $\iint dydx$ on the domain $0\leq r\leq1$, ${\pi}/{3}\leq\theta \leq{2\pi}/{3}$ [duplicate]

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\bigg\{\frac{\pi}{3}\leq\theta \leq\frac{2\pi}{3}\bigg\}\;\;\;\;,0\leq r\leq1$ $\color{blue}{\text{without using polar ...
1
vote
4answers
46 views

Finding the equation of the straight line $y=ax+b$?

If I have a circle $x^2+y^2=1$ and line that passes trough $(0,0)$ and I know the angle between the line and the axis. If, for example, the angle is $\frac{\pi}{3}$, how can I find the equation of ...
-1
votes
2answers
39 views

Finding matrix representation of an Ellipsoid [on hold]

I have a $2$-dimensional ellipsoid centered at $(1,2)$. The axes are parallel to $y=x$ and $y=-x$, and it passes through points $(-1,0)$, $(3,4)$,$(0,3)$,$(2,1)$. I would like to find the symmetric ...
0
votes
0answers
26 views

Rotating one coordinate system about another

I have two coordinate systems: A and B. I also have a point p, whose position relative to ...
3
votes
0answers
52 views

What is the name of this shape? (spacetime)

After seemingly endless searching for terms such as curved cone, hyper-cone etc I am at a loss as to what this shape is called. I believe it is commonly used to depict the curvature of space time. ...
0
votes
1answer
25 views

Clarify formula that computes number of dies on wafer

I want to compute the number of dies per wafer (also DPW in the following). There are some formulas, that can be used to do so: ...
1
vote
1answer
41 views

A good book on basic (Euclidean) geometry.

We were studying demonstrative geometry, so I thought if I read Euclid's Elements it would give me the proper conceptual basis to understand the theorems. But then I learned that Euclid's method of ...
1
vote
0answers
32 views

Number of triangles possible in android lock patterns?

I recently starting using the patternlock on my android phone and i play around with it a lot, just drawing lines until im locked out for 30 secs. I thought i'd make it into a pointless game of ...
0
votes
3answers
43 views

Angle between medians in right triangle

In a right angled triangle,medians are drawn from the acute-angles to the opposite sides.If maximum acute angle between these medians can be expressed as $tan^{-1}(\frac{p}{q})$ where p and q are ...
0
votes
2answers
48 views

Farthest point on parallelogram lattice

On points arranged in a parallelogram lattice, like on the image in this Wikipedia article, how to calculate the maximal distance any point on the plane may have to its closest point from the lattice. ...
0
votes
1answer
25 views

How to show the average x-coordinates of four collinear points on the curve is a constant?

Show that if four distinct points of the curve $y=2x^4+7x^3+3x-5$ are collinear then their average x-coordinate is some constant k. Find k. Shall I use vector to calculate their x-coordinate, or ...
3
votes
4answers
77 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
1
vote
0answers
46 views

“Maximum point lies on a curve” implies tangential derivative is zero there.

Given a differentiable function $f:\mathbb{R}^2\to\mathbb{R}$, suppose that it has a local maximum at the point $(x_0,y_0)$. Let $\gamma$ be a smooth curve passing through $(x_0,y_0)$. Does it follow ...
2
votes
2answers
30 views

Using the cross product to find the angle between two vectors in $\Bbb R^3$

Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle$$. Find the angle between $u$ and $v$, first by using the dot product and then using the cross product. ...
0
votes
0answers
20 views

Prove winding number is the same as index of a vector field.

I'm trying to prove that the winding number and the index around a point in a vector field are the same. I know that the index is sometimes defined as the winding number but I'm working with the ...
2
votes
2answers
31 views

Parallelogram ABCD

There's a parallelogram $ABCD$. I'm given point $A(3,12)$ and point $B(-1,5)$. Given the equations of the lines $BC$ and $AC$ are $y=8x+13$ and $y=3x+3$ respectively. How to find the coordinates of ...