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

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1
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3answers
30 views

Show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$

Show that in a $\Delta ABC$, $\sin\frac{A}{2}\leq\frac{a}{b+c}$ Hence or otherwise show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$ for all $n\geq1$. ...
2
votes
2answers
26 views

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$ How should i attempt this question?I thought over it hard but could not ...
0
votes
0answers
24 views

What is the name of a cilinder-like object based on an ellipse instead of a circle?

If I project a circle over the Z-axis, I'll get a cilinder. If I project a square over the Z-axis, I'll get a parallelepiped. If I project an ellipse over the Z-axis, I'll get a... whatsitsname? I ...
2
votes
0answers
28 views

Geometry/Trigonometry Determine angle in a Triangle

Triangle ABC is isosceles with BC as base, AB=AC and Angle A=20 degrees. Points D and E lie on sides AB and AC respectively, such that D lies between A and B, E lies between A and C, angle BCD=50 ...
1
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0answers
13 views

Jacobi determinant for high-dimensional sphere inversion

I need to find the Jacobi determinant for the unit sphere inversion in $\mathbb{R}^n$, i.e. the map given by $f(x) = \frac x {|x|^2}$ for $x\in \mathbb{R}^n$. The main problem is to figure out the ...
1
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1answer
11 views

determine cube orientation given one side in a perspective projection

Suppose that we are given an arbitrary quadrilateral T that does not have any parallel edges. I want to draw a cube in a three-point perspective projection such that T is one of its sides. The ...
3
votes
4answers
35 views

$R$ is the midpoint of $MN$ and the points where $AC$ intersects $MD$ and $ND$ are $P$ and $Q$, respectively. Show that $PR=QR$.

In square $ABCD$, $M$ and $N$ are points on $AB$ and $BC$, respectively such that $\angle MDN=45°$. $R$ is the midpoint of $MN$ and the points where $AC$ intersects $MD$ and $ND$ are $P$ and $Q$, ...
3
votes
0answers
45 views

Very special geometric shape (No name yet?)

I suppose this geometric shape is something very 'special'. I cannot clarify in short about being 'special', but I think this shape stands together with such special shapes like square and hexagon. ...
0
votes
0answers
28 views

Trignometry, bearings yacht race question [on hold]

In a yacht race each yacht has to sail around a set of 4 buoys, and then return to the start line in order to finish. We will assume that the buoys are just points and the start line is also a ...
0
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0answers
15 views

book suggestions on homogeneuos coordinate and projective geometry [on hold]

Will any one introduce some good books about projective geometry and homogeneuos coordinates to me?
2
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0answers
10 views

About the interior of a polytope

Let us consider a polytope in $\mathbb{R}^n$ (in this context it must NOT be bounded) $\mathcal{P} = \{ x: A \cdot x \leq c\}$ for some matrix $A$. Let $\mathcal{I} = \{ x: A \cdot x < c\}$ be a ...
-1
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0answers
22 views

Straight lines and its applicaton

What are the number of lines passing through (1,1) and intersecting a segment of length 2 unit between the lines x+y=1 and x+y=3 ?
0
votes
3answers
38 views

Equilateral Triangle Property

If the vertices of a triangle have integral coordinates how to prove that the triangle cannot be equilateral ?
0
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1answer
20 views

How to describe the transformation that changes French flag to Russian flag?

http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=Russia%2C%20France%20flags I presume it can be described two group operators, but I'm not sure how to come up with the formal description. ...
0
votes
1answer
11 views

Find the cost of leveling the triangular plot.Find the area using Herons formula

A Triangular Plot has sides $600$m .$640$m and $700$m.How much would the leveling cost if $1m^2=50$Rs. Solve the Problem using Heron's Formula.
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1answer
15 views

Determining North-South Line Via Non-digital Watch Method: Discussion on Background Theory [on hold]

Read this recently (page 9). States that if you point the current hour hand at the sun, then the angle bisection between it and an imaginary line running through the 12 hour position will point south. ...
0
votes
2answers
15 views

confused on to leave in centimeters or convert to cubic centimeters

The volume $V$ of the cylinder is $65\pi \mathrm{cm}^3$. The height of the cylinder is $5$ centimeters. Use the formula $V = Bh$ to find the area of the base of the cylinder.
0
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0answers
13 views

What are the loci of points z which satisfy the following relations?

a) |Z-Z1|=|Z-Z2| b) 0< Re(iZ)<1 c) |z|=ReZ+1 d) Im((Z-Z1)/(Z-Z2))=0 The professor did not explain loci in class and the text does not have any examples, so I am completely lost.
-1
votes
1answer
25 views

Is there any practical use of $0^\circ angles$?

An angle is defined as two rays $\overrightarrow{YX}$, $\overrightarrow{YZ}$ sharing a midpoint $Y$; the angle formed is called $\angle XYZ$ or simply $\angle Y$. Then, the measure of $\angle Y$ is ...
-2
votes
1answer
33 views

Proving the AIP theorem?#2

How do you prove the AIP theorem? (if a pair of alternate interior angles formed by a transversal intersecting two lines m and n are congruent then line m is parallel to line n) I already know you can ...
0
votes
1answer
24 views

The Length of an edge of the cube

One dimension of a cube is increased by 1 inch to form a rectangular block. Suppose that the volume of the new block is 150 cubic inches, find the length of an edge of the original cube.
3
votes
2answers
36 views

Deriving the formula of the Surface area of a sphere

My young son was asked to derive the surface area of a sphere using pure algebra. He could not get to the right formula but it seems that his reasoning is right. Please tell me what's wrong with his ...
0
votes
4answers
47 views

A question on finding for the dimensions of a rectangle [on hold]

The diagonals of a rectangle is 8 m longer than it‘s shorter side. If the area of the rectangle is 60 square meters, find its dimensions.
0
votes
0answers
18 views

staircase length in Whitney's flat norm and Jenny Harrison's natural norm

Can someone provide the complete calculation for the length of a staircase as it converges to a diagonal line in Euclidean space in a sequence in which the number of steps goes to infinity between two ...
0
votes
0answers
16 views

Compact set in the interior of a cone

Suppose compact set $S \subseteq R^n$ is in the interior of $x_0+C$, where $C$ denotes a solid convex cone in $R^n$ with the vertex at $0$. I am trying to prove that $\exists r>0$ such that $$S ...
1
vote
3answers
21 views

Finding a coordinate over a right angle in a triangle where the other two coordinates are known

a B ------- C \ | \ | \ | c \ | b \ | \ | \| A Alright, this is a triangle I have, and ...
0
votes
2answers
22 views

General equation of line that goes through center of a circle and a point

Given an arbitrary point $P$, at $(x_{1}, y_{1})$, is there a general expression of a line that goes through a circle of radius $r$ centered at the origin? I know there are infinite number of such ...
1
vote
1answer
26 views

I have a problem in loci [on hold]

$ABCD$ is a square with side length $4$ cm. A variable point $P$ moves inside the square so that $PA\leq 4$ cm, $PC\leq PA$ and the area of $ABP$ is $\leq 6$ cm$^2$. Construct $ABCD$ accurately and ...
1
vote
1answer
29 views

A geometry question on finding the area of cyclic quadrilaterals

The circumcircle of a cyclic quadrilateral $ABCD$ has radius $2$. $AC, BD$ meet at $E$ such that $AE = EC$. If $AB^2 = 2\cdot AE^2$ and $BD^2 = 12$, what is the area of the quadrilateral?
1
vote
1answer
30 views

Validity of my proof by contradiction of converse of Pythagorean Theorem.

So in, $\triangle ABC$ it is given that $AB^2=AC^2+BC^2$. Let us assume that $\angle C\neq{90}^{\circ}$. And let us make a perpendicular $AD$ to $BC$. Now, by the Pythagorean Theorem, in $\triangle ...
2
votes
2answers
33 views

When do circles on the sides of a triangle intersect?

If you have a non degenerate triangle and two of the sides are chords of two respective circles, then under what conditions do the circles intersect at two distinct points? I'm having trouble ...
0
votes
0answers
32 views

Is math, in the end, only geometry [on hold]

When thinking about the Universe, or "reality", Isn't every part of mathematics a tool for expressing something geometrically further down the line? Yes, every part of math is related, but isn't ...
1
vote
1answer
20 views

Tool for the partition problem with planar rectangles

The classical "partition problem" asks how many ways one can write a given natural number as a sum of smaller numbers. One variant of this would be to ask if a positive real number can be expressed ...
0
votes
1answer
22 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
2
votes
4answers
98 views

Geometric proof of $QM \ge AM$

Prove by geometric reasoning that: $$\sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}$$ The proof should be different than one well known from Wikipedia: DISCLAIMER: I think I devised such proof ...
1
vote
1answer
22 views

Finding the distance from a mirror

My friend sent me this problem, and my efforts to solve it have thus far been frustrated. I need some insight! Joe is 6 feet tall, and standing in front of a mirror that is at eye level, and is 3 ...
2
votes
1answer
13 views

Lines and projective isomorphism

Let $\rho : P(\mathbb{R^3}) \to P(\mathbb{R^3})$ be an homography, this is, a projective isomorphism induced from the isomorphism of vector spaces. I'm trying to understand what information about ...
1
vote
0answers
33 views

Frechet mean of $k$ elements in the n-dimentional sphere.

The Frechet mean of $k$ elements $x_1, \ldots, x_k \in S = \{ x \in \mathbb{R}^n,\, \| x \| = 1 \}$ is defined as the $\text{arg}\underset{\|x\|=1}{\text{min}} \sum_{i=1}^n d^2(x_i,x)$. Where ...
1
vote
1answer
58 views

How to calculate distance from the International Space Station given coordinates?

How would one calculate how far away a point is (latitude/longitude) from the international space station given its latitude/longitude/altitude? The distance would be direct as if drawing a straight ...
1
vote
0answers
43 views

Proof of Simple Properties of Volume

Let $e_{1},\ldots,e_{n}$ be vectors in $\mathbb{R}^{n}$. Define a parallelepiped $P$ to be a translate of the set $$\left\{x\in\mathbb{R}^{n} : x=t^{1}e_{1}+\cdots+t^{n}e_{n}, 0\leq t^{i}\leq ...
1
vote
1answer
49 views

Getting the angle that is needed for covering a given distance on an ellipse's cirumference

In a small programming exercise I asked myself, I want to calculate various things about ellipses. The part I'm stuck with is the following: I want to calculate the angle that is needed cor covering a ...
0
votes
1answer
32 views

Show that a point lies on the diagonal of quadrilateral

In a quadrilateral ABCD we choose a point E on the side AD and a point F on the side CD. Then we choose a point G on the line EF. Let H be the second point of the intersection between the circles that ...
1
vote
1answer
21 views

geometrical problem on triangle [on hold]

Given a triangle ABC, if $$a = \dfrac{2(b^2 - c^2)}{-b + \sqrt{b^2 + 4c^2}}$$, prove that $3m(C) = 2m(B)$.
0
votes
1answer
20 views

Write $CX,AY,BZ$ in terms of $CA,CB$ and the ratios $\alpha, \beta, \gamma$?

The point $X$ divides $AB$ in the ratio $\alpha$, $Y$ divides $BC$ in the ratio $\beta$ and $Z$ divides $CA$ in the ratio $\gamma$. Write $CX,AY,BZ$ in terms of $CA,CB,\alpha, \beta, \gamma$. I ...
4
votes
1answer
36 views

Combinatorial problem of choosing points inside an equilateral triangle without them being too close.

Determine the smallest integer $m_n$ which satisfies the following property: If $m_n$ points are chosen inside an equilateral triangle of sides 1, then at least two of them are at distance ...
0
votes
1answer
17 views

What's the necessary condition for that any three vectors are parallel to the edges of a triangle in the plane?

What's the necessary condition for that any three vectors are parallel to the edges of a triangle in the plane? I answered the following: The necessary condition is that the vectors are ...
0
votes
1answer
14 views

How do I find a Bézier curve that fulfills a given width and height?

I am building a software application that works with vector graphics and I need to use Bézier curves to draw a heart shape, like this one here which I created in MS Paint: The only information ...
0
votes
1answer
17 views

Geometric proof of circular mil formula $A=d^2$

The circular mil "is a unit of area, equal to the area of a circle with a diameter of one mil." "The area in circular mils, A, of a circle with a diameter of d mils, is given by the formula:" ...
1
vote
1answer
31 views

Width of trapezoid at any height?

Assuming I have a trapezoid where I know the height, bases, and legs, I would like to obtain the width of this trapezoid at any height y. What I want is very similar to the median formula for a ...
1
vote
1answer
36 views

Metric tensor for n-sphere in ambient coordinates

Let $S^n$ be the unit n-sphere embedded in $\mathbb{R}^{n+1}$: $$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$ What is the induced metric tensor for the sphere, in $\mathbb{R}^{n+1}$ ...