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

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Dihedral angle Finding

I meet a wall while some problem solving. The wall is following question. There is a triangle ABC, The vertice A touch bottom plane and the distance from B, C to the bottom : BE = b, CD = c . When ...
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0answers
22 views

How would I solve this question?

Let ABC be an acute angled triangle. The arc between A and B of the circumcircle of ABC is reflected through the line AB, and the arc between A and C of the circumcircle of ABC is reflected over the ...
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0answers
14 views

Tipi Shelter Formula

Autumn is quickly approaching in my neck of the woods and with it brings the start of camping and backpacking season. This year I want to work on my bushcraft skills with one particular project in ...
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0answers
39 views

Are there more ways to divide the “L”-shaped tromino into four congruent parts?

Recently my sister-in-law, who is training to become a high school mathematics teacher, asked me the following question: Consider the following polygon constructed by adjoining three squares of ...
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1answer
41 views

Prove a closed ball is a subset of another ball iff the triangle inequality is true

For balls in $\Bbb{R}^n$ prove that: $$\bar{B}(a,r) \subset B(b,s) \iff \|a-b\| \lt s - r.$$
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4answers
26 views

How can I find the length to this geometry problem?

A person 6 feet tall is standing at the base of a lamp post that is 25 feet tall and then begins to walk away from the lamp post. When the person is 10 feet from the lamp post, what is the length of ...
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1answer
21 views

A question about angles in the Euclidean plane

It has long been known that an arbitrary angle (in the Euclidean plane) cannot be trisected using only ruler and compass, but that this can be done using a mechanical linkage. Given any positive ...
3
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1answer
20 views

Rolling a circle around a two dimensional curve

This is a sort of funny idea I had the other day, and although I expect to get a very technical answer I am fine with any intuitive explanation. Consider being given a function in the plane, for ...
3
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1answer
22 views

A strange property of continuous deformations of balls

Let $B$ be the closed unit ball in $\Bbb R^n$ and let $$F:B\times[0,\infty)\to\Bbb R^n,\quad F(x,t)=F_t(x)$$ be a continuous map such that $F_0$ is the identity. In other words, $F$ defines a ...
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0answers
19 views

Let $M_1 , M_2$ two manifolds of dimension $n_1, n_2$ and $M_1 \subset M_2$. Prove: $M_1$ is relatively open in $M_2$ $\iff n_1 = n_2 $.

Let $M_1 , M_2$ two manifolds of dimension $n_1, n_2$ and $M_1 \subset M_2$. Prove: $M_1$ is relatively open in $M_2$ $\iff n_1 = n_2 $. I have no idea where to start with this one, any help would be ...
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2answers
21 views

What is the graph that corresponds to $Q'_8$ generalized quadrangle ? Could you please explain this in plain english?

In this paper a table about large graphs with given degree and diameter graphs is shown: I would like to know what the adjacency list of the graph denoted by: in the table above is. Could ...
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2answers
28 views

Lines and planes - general concepts

I've come across a book that has this general questions about lines and planes. I can't agree with some of the answers it presents, for the reasons that I'll state below: True or False: Three ...
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2answers
31 views

Proving co-ordinates of an equilateral triangle are integers in a plane

Consider the 2D plane $P$ in $\Bbb{R}^3$ defined by $$P=\{x \in \Bbb{R}^3 \mid x_1+x_2+x_3=0\}.$$ Let $a$, $b$, $c$ be the vertices of an arbitrary equilateral triangle in $P$ such that all the ...
3
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0answers
38 views

Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
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2answers
33 views

If in a triangle $ABC$,$1=2\cos A\cos B\cos C+\cos A\cos B+\cos B\cos C+\cos C\cos A$,then prove that triangle will be equilateral triangle

If in a triangle $ABC$ we have $$1=2\cos A\cos B\cos C+\cos A\cos B+\cos B\cos C+\cos C\cos A\ ,$$ then the triangle will be equilateral triangle. I tried but except few steps,could not prove it. ...
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2answers
18 views

volume of a $n$-d parallelepiped with sides given by the row vectors of a matrix $A$ is the product of the singular values of this matrix $A$

Why the volume of a $n$-dimensional parallelepiped with sides given by the row vectors of a matrix $A$ can be seen as the product of the singular values of this matrix $A$? I only know in ...
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1answer
58 views

Prove that $\frac{r_1}{r-r_1}+\frac{r_2}{r-r_2}+\frac{r_3}{r-r_3}=\frac{r_1r_2r_3}{(r-r_1)(r-r_2)(r-r_3)}$ [on hold]

Let $D,E,F$ be the feet of the perpendiculars from the incenter $I$ to the sides $BC,CA$ and $AB$ respectively. If $r,r_1,r_2$ and $r_3$ are the inradius of the triangle $ABC$ and radii of the circles ...
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0answers
32 views

Show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$

If the internal bisectors of the angles of the triangle ABC make angles $\alpha,\beta,\gamma$ with sides $a,b,c$ respectively then show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$ I tried to ...
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1answer
15 views

Given the end-vertices of two line segments, how do you calculate the point at which they intersect?

Given only the vertices of each line segment, and it's assumed they intersect, how do I calculate the point at which they intersect (in two and additionally three dimensions)?
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1answer
26 views

Euclidean Geometry (Potential Menelaus Theorem)

I have a strong suspicion that this problem applies Menelaus's theorem, but I can't see it. I also tried algebraic manipulation (such as trying to re-write BD/DC in terms of AB or CP), but to no ...
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2answers
45 views

proof of isosceles triangle?

How do you prove this isosceles triangle? Given line AC is congruent to line BC Prove: Angle A=Angle B I've gotten to the angle bisector and SAS(side- angle- side), and I believe there is one more ...
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2answers
25 views

Smallest convex polyhedron containing integer points of a cylinder

A cylinder has height $6$ and radius $3$. The centers of the two bases are $(0,0,0)$ and $(0,0,6)$. Find the volume of the smallest convex polyhedron that encloses every lattice point inside the ...
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2answers
42 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
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2answers
36 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 ...
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0answers
25 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
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0answers
31 views

Geometry/Trigonometry Determine angle in a Triangle [duplicate]

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 ...
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0answers
17 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 ...
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1answer
17 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 ...
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4answers
43 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
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0answers
54 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. ...
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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 ...
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0answers
17 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?
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0answers
19 views

About the interior of a polyhedron

Let us consider a polyhedron 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 ...
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23 views

Straight lines and its applicaton [on hold]

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 ?
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2answers
43 views

Equilateral Triangle Property

If the vertices of a triangle have integral coordinates how to prove that the triangle cannot be equilateral ?
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1answer
21 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. ...
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1answer
12 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
44 views

Determining North-South Line Via Watch Method: Theory & Reason

I recently read that if you're in the northern hemisphere and have an analog watch, then you can point the hour hand at the sun and know that a south line lies between (bisection) the hour hand and ...
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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.
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18 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.
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1answer
28 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 ...
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1answer
43 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 ...
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1answer
25 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.
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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 ...
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4answers
48 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.
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0answers
26 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 ...
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0answers
33 views

Does a compact set in the interior of a cone also belong to the intersection of all slightly perturbed cones?

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 ...
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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 ...
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2answers
23 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
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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 ...