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

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1answer
13 views

A sheet of cardboard measures 15cm by 7cm. Four equal squares are cut out of the corners and the sides are turned up to form an open box. [on hold]

a) If the edge of the cut out square is x cm, express the dimensions of the box in terms of x. b) What are the restrictions placed on the values of x? (i.e. the implied maximal domain).
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1answer
25 views

Quadratic forms and midpoints

The midpoint of the vectors $u$ and $v$ is $w=\frac{u+v}{2}$. In euclidean geometry, an alternative characteristic of midpoints is $|v-w|=|u-w|=\frac{1}{2}|u-v|$. I wonder if this generalizes to ...
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2answers
50 views

Compute the integral over the volume of a torus,

In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., $$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$ Let $\Omega$ ...
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1answer
26 views

Line segment equation in polar coordinates

I have a line segment given by two points $A$ and $B$. $$A+u(B-A), u\in[0,1]$$ when doing calculations with this segment, it would be advantageous to have it written in polar coordinates around some ...
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3answers
28 views

Slope of a line segment.

If $A(x_1, y_1)$ and $B(x_2, y_2)$, we know that slope $m = \frac {(y_2 - y_1)} {(x_2 - x_1)}$. What decision can we take aout the line segment when, $m = \frac 0 0$, $m = \frac {dy} 0$, and, $m = ...
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0answers
20 views

Find the constant $k$

In a graph $G$ with $n$ vertices, let $T_1$ be the number of triangles one can make and $T_2$ be the numbers of tetrahedrons. Find the least constant $k$ such that $(T_2)^3\le k.(T_1)^4$.
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3answers
28 views

Question about concyclic points on the coordinate axes

If the points where the lines $3x-2y-12=0$ and $x+ky+3=0$ intersect both the coordinate axes are concyclic,then the number of possible real values of k is (A)1 ...
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2answers
38 views

External bisectors of the angles of ABC triangle form a triangle $A_1B_1C_1$ and so on

If the external bisectors of the angles of the triangle ABC form a triangle $A_1B_1C_1$,if the external bisectors of the angles of the triangle $A_1B_1C_1$ form a triangle $A_2B_2C_2$,and so on,show ...
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0answers
63 views

Complex numbers: conjugate [on hold]

Can anyone help me with this? Let $z^*$ be the complex conjugate of $z$ (a) Show that $ (zw)^* = z^*w^*$ (b) Prove by induction that $\ (z^n)^*=(z^*)^n$ for all positive integers $n$.
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4answers
80 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
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3answers
24 views

Finding the equation of a circle given 3 points without using elimination [on hold]

find the equation of the circle using points (-4,-4) (-3,1) (2,0) without using elimination.
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4answers
43 views

If a triangle has 2 sides of equal length, is it isosceles?

I know that if the 2 angles of a triangle are the same, it is isosceles. But what if the two sides are the same? Can we conclude that the corresponding angles are the same and it is isosceles?
2
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1answer
22 views

Creating Gosper Curve by geometry

I am reading a book by "Fractals, Chaos and Power Laws" by Manfred Schroeder.On page 13, it produces seven fractal tiles from seven hexagons by breaking up each side into a three-piece zigzag as shown ...
2
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0answers
48 views

IMC 2014, Problem 4 [Day 2]

We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points ...
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0answers
18 views

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given? [on hold]

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given as ( x1, y1 ) ? where T = x(x1)/aa + y(y1)/bb - 1 and S1 = (x1)(x1)/aa + (y1)(y1)/bb - 1 where 2a ...
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1answer
105 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
39 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 ?
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1answer
38 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
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1answer
34 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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1answer
59 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 $O$ be an arbitrary point located inside the triangle $ABC$. Let $D, E, F$ be (respectively) the projections of $O$ on $BC, CA, AB$. Prove that $$\cot{\angle ADB} + \cot{\angle BEC} + ...
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0answers
33 views

Geometry problem… [on hold]

GH=OP Find $x$ [1 any help guys ??
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1answer
38 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 ...
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2answers
19 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.
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2answers
34 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: $\dfrac{y-b}{r}=\dfrac{y}{s}$
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1answer
34 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.
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2answers
20 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
1answer
59 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 ...
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0answers
20 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 ...
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2answers
52 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 ...
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0answers
53 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
45 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 ...
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4answers
87 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 ...
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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 ...
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1answer
16 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} ...
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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
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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 ...
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3answers
54 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
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1answer
38 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
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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 ...
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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 ...
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2answers
41 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? ...
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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
29 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
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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
29 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 ...
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0answers
40 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 ...
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0answers
10 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.
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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 ...