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
51 views

Given an acute triangle ABC with altitudes AH, BK. Let M be the midpoint of AB

Given an acute triangle ABC with altitudes AH, BK. Let M be the midpoint of AB. The line through CM intersect HK at D. Draw AL perpendicular to BD at L. Prove that the circle containing C, K and L is ...
2
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1answer
84 views

Is the area of this pentagon $4-\sqrt 5$?

Consider a regular pentagon with vertices (in clockwise order) $A, B, C, D, E$, let $A'$ be the point of intersection of $BD$ and $CE$, let $B'$ be the point of intersection of $CE$ and $DA$, and ...
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2answers
22 views

Similarity conditions of two right trapezoid with similar angles

We have $2$ right trapezoid for example two trapezoid with angles $90^{\circ},90^{\circ},80^{\circ},100^{\circ}$. do we need to all the sides proportionality or less is enough ?
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2answers
60 views

Is the Dikin Ellipsoid actually a ball?

I have the inequality (row wise): $Ax \leq b$ The Dikin ellipsoid centered at $x_0$ with radius $r$ is: $$\{z \quad | \quad (z-x_0)^T(z-x_0) \leq \frac{r^2}{H(x_0)}\}$$ where, $$H(x_0) = \sum ...
0
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0answers
25 views

Find distance between a plane and some points [closed]

Consider points $x_1,\ldots,x_n$ and plane $w\cdot x-\gamma=0$ in $\mathbb{R}^n$ and let $A=[x_1,x_2,\ldots,x_n]^T$. Is correct following formula to find the distance between these points and the ...
2
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2answers
58 views

Question based on triangle inscribed in unit circle

$ \bigtriangleup ABC $is inscribed in a unit circle.If angle bisectors of internal angles at A,B and C meet the circle at D,E and F respectively then value of $\frac{AD \cos\frac{A}{2}+BE ...
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1answer
24 views

Intersection of Cut Locuses

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then when is: \begin{equation} \bigcap_{p\in M} C_p(M)=\emptyset\text{ ?} \end{equation}
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1answer
87 views

Prove that $\tan\alpha =\tan^{2}\frac{A}{2}.\tan\frac{B-C}{2}$

Given a triangle ABC with the sides $AB < AC$ and $AM, AD$ respectively median and bisector of angle $A$. Let $\angle MAD = \alpha$. Prove that $$\tan\alpha =\tan^{2}\frac{A}{2}\cdot ...
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0answers
19 views

Find the best trapezoidal fitted in an irregular shape [closed]

I am working with some earth irrigation canals. Irrigation canals are usually in trapezoidal shape. These trapezoidal canals are defined by the width of bottom of canal (B) and high of depth of canal ...
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3answers
73 views

what is the definition of cosine , sine [duplicate]

I know that sine is the ratio of the perpendicular to the hypotenuse of an acute angle. Similarly cosine is the ratio of the base and hypotenuse . But now I found that there is sine and cosine of an ...
4
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0answers
59 views

How is it that $\pi$ appears in so many formulas that seem to be in no way geometric. [duplicate]

When I first saw: $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\mp\cdots.$$ I was puzzled that such an expression could have anything to do with circles. There are tons more and ...
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2answers
44 views

Find a plane with distance $3$ from $3x-y-z = 0$

I need to find a plane such that its distance from the plane $3x-y-z = 0$ is $3$. Since distance is defined only for parallel planes, I already know that they have to be parallel, and then, the ...
0
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0answers
26 views

Is there a theory for cellular automata propagating signals in straight lines?

Is there a theory explaining how a cellular automata can propagate signals in straight lines? For example, this video shows how some "signals" travel down at a diagonal, even though they are composed ...
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0answers
44 views

Beyond Pythagoras [closed]

Draw an arbitrary triangle $\triangle ABC$. Measure its sides. Draw a ray, $BC$. Draw a circle with radius $AB$. Find the point of intersection, $D$. Measure segment $\overline{CD}$. ...
2
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2answers
67 views

Triangle Geometry and Circles Problem

I have discovered something using Geogebra and I am positive it is true. I have tried to prove and my solution works but it is extremley convoluted. I'm hoping someone can provide a simple proof of ...
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5answers
44 views

co-ordinate geometry question 3 [closed]

Find intercept of the line whose intercept of $x$-axis and $y$-axis are respectively twice and thrice of those by the line $3x+4y=12$ ?
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0answers
23 views

Symmetry of stereographic projections of tangent vector to $S^2$ at equator

There is a vector lying in the tangent plane to a sphere $S^2$ at equator. We take two its "stereographic" projections - one from the south pole and other - from north. Projections to the planes ...
0
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1answer
36 views

Number of triangles having particular area

If $g:R\to \ N\cup\big\{0\big\}$ and $g(x)=n$,where $x$ represents the area of triangle joining the two fixed points and a variable point $R(p,q)$such that $\angle PRQ=\frac{\pi}{2}$ and $n$ ...
0
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1answer
11 views

Finding $x$ coordinates on a rectangle if Rectangle $a$ was scaled up to Rectangle $b$

I wasn't too sure how to explain the question in the title so i drew up my problem that I am trying to solve: http://i.imgur.com/PyWMh6f.jpg Basically I choose a point on rectangle $A$ and then find ...
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2answers
27 views

A triangle and its median in complex plane.

Let $z_1$, $z_2$, $z_3$ be vertices of the triangle $\triangle ABC$. And given that $|z_1|=|z_2|=|z_3|$. Then the median through $A$ cuts the circumcircle at which point? We need to get the answer in ...
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0answers
33 views

Should I try to figure out geometric construction?

I've been reading and working through the book - Euclid and Beyond by Robin Hartshorne. In the first chapter there are a few constructions that I haven't been taught in school (just finished 11th ...
4
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3answers
201 views

Finding the area of a square that has a circle inside itself

I tried to solve the following problem: I think the image is self-descriptive. I tried to draw a vertical line from the top-end of $\theta$ angle to the horizontal line, then tried to use the ...
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3answers
34 views

Find more integral points on a hyperbola

Let $\mathcal H$ be a hyperbola (in the affine plane) whose defining equation has integers coefficients. Assume that one knows 2 points of $\mathcal H$ with integral coordinates. Is there a way to ...
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2answers
50 views

Interesting problem in congruence of triangles

While solving the exercises of my book I came across this interesting problem: $\triangle ABC$ is isosceles triangle with $AB=AC$. D is a point on base BC such that $AD$ perpendicular on $BC$. To ...
1
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1answer
43 views

Is there a specific mathematical term for a shape whose dimensions are defined?

When I say the word "circle", I know that I have described a "shape". Specifically, a "circle" is the shape formed by the set of all points in a plane that are at a given distance from a given point. ...
11
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0answers
97 views

Penrose's remark on impossible figures

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...
1
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1answer
65 views

Finding the intersection points of common tangents on a pair of non-intersecting ellipses

I'm having some trouble with this, I don't know why but for some reason it is giving me a lot of trouble. Ultimately I intend to implement it into a program for modelling something, but I cannot even ...
1
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1answer
44 views

Determining the position of a polygon inside a circle from only the angle of opposing sides/edges.

For illustration click here I have a simple convex irregular polygon (octagon in example image) inside a circle (circle and polygon are not always concentric and never touching or intersecting) and I ...
2
votes
1answer
62 views

Folding a paper such that the size of one sides be as minimum as possible?

Suppose that we have an A4 paper like this: How to fold this paper such that the bottom-right corner overlap the left edge of the paper and that the size of AB side be as minimum as possible. It ...
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2answers
32 views

Polygon and line intersection

Does anyone help me with the fast algorithm to determine the intersection of a polygon (rotated rectangle) and a line (definite by 2 points)? The only true/false result is needed.
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3answers
65 views

Sphere packing question

I'm a secondary school maths teacher, currently on my holidays working through some maths problems for fun. Here is one I have done, but it felt too easy, so if you could check if there's any ...
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2answers
51 views

What is the precise name of this quadrilateral?

Picture I was thinking about half rectangle but there must be better name for it. You can assume r1r2 and r3r4 are not equal. What is the name of this geometric shape?
2
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0answers
41 views

Deriving trigonometric derivatives in a purely geometric manner [duplicate]

Is there any geometric way to derive basic trigonometric derivatives (without using the cos/sin of the sum formula)? For example: $\dfrac d {dx}\cos x=-\sin x$
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4answers
42 views

What is the equation of a 3D line which represents the intersection between two 3D planes?

The intersection defined by the two planes $v \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$ and $v \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$ is a line. What is the equation ...
1
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1answer
32 views

Distance between two 3D lines

What is the distance between the 3D lines $x = \begin{pmatrix} 1 \\ 2 \\ -4 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \\ -1 \end{pmatrix} t$ and $y = \begin{pmatrix} 0 \\ 3 \\ 5 \end{pmatrix} + ...
3
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0answers
32 views

'Unrolling' the neighbourhood of a space curve

I have a space curve $\gamma : \mathbb{R} \longrightarrow \mathbb{R}^3$, sampled at $n$ discrete points. I have implemented an algorithm that gives me an approximation to $\gamma$'s tangent, normal ...
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3answers
64 views

Find the side of an equilateral triangle inscribed in a circle.

Excuse the poor drawing. $\triangle CDE$ is an equilateral triangle inscribed inside a circle, with side length $16$. Let $F$ be the midpoint of $DE$. Points $G$ and $H$ are on the circle so ...
4
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1answer
42 views

A question about a curve on the surface of a sphere

Let the three points A,B,C be the vertices of a moving spherical triangle on the surface of a sphere. The triangle moves so that while the vertices A,B remain fixed, the angle BCA at the vertex C ...
11
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6answers
564 views

Wanted : for more formulas to find the area of a triangle?

I know some formulas to find a triangle's area, like the ones below. Is there any reference containing most triangle area formulas? If you know more, please add them as an answer ...
1
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1answer
26 views

Equation to place points equidistantly on an Archimedian Spiral using arc-length

I am looking for a way to place points equidistantly along an Archimedes spiral according to arch-length (or an approximation) given the following parameters: Max Radius, Fixed distance between the ...
0
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1answer
20 views

Finding the Aspect Ratio for Stacked Boxes

I am trying to figure out how to calculate the ideal aspect ratio for a collection of boxes. All of the boxes always maintain the same aspect ratio, but can be scaled to fit the following layouts: ...
1
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1answer
26 views

Geometric interpretation of the derivative of a Bezier curve

For a given set of control points $b_0, b_1, \ldots, b_n$, the Bezier curve is defined as $$b^n(t) := \sum_{j=0}^n b_j B_j^n(t),$$ where $B_j^n(t):=\binom{n}{j}t^i(1-t)^{n-i}$ are Bernstein ...
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1answer
55 views

What is the difference between a rigid motion and an isometry?

Can those two terms be used interchangeably?
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1answer
23 views

Findings the dot product between two non-adjacent vectors

I need to find $r\circ(p-q)$ from the below diagram, and since $r$ is prependicular to $p$, I only need to calculate $-(r\circ q)$ when I know that the modulus of $q=3$ and of $p=4$. Now, I know that ...
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1answer
27 views

Proving that $R^2$ with Euclidean metric is isometric with $R^2$ with maximum metric?

I am reading a geometry book on my own and can't figure out how to prove it. I cannot figure out a transformation that preserves the distances for ALL points. The Euclidean metric is $$d_1(A,B) = ...
0
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1answer
32 views

Determining the fourth vertex of a parallelogram knowing that its the point of intersection of two circles

This question was part of the exercises in one of the courses i'm taking. The answer was already provided. The first circle was assumed to have as its center, vector $v_1$, while its radius was the ...
2
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1answer
36 views

What is the number of distinct elements in $S$?

Allow for these values: $$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix} \text{ and } B = \begin{pmatrix} 1 ...
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0answers
24 views

Characteristic polynomials for matrix A, involving the Identity matrix

Let us say we have a square matrix A, where A's characteristic polynomial is defined as $P_A(t) = \det (t I - A)$ (In this problem, I represents the identity matrix which has the same dimensions as ...
1
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1answer
98 views

Find A such that $A^2 \neq I$ but $A^4 = I$ [duplicate]

Find a $3 \times 3$ matrix A such that $A^2 \neq I$ but $A^4 = I$, where $I$ is the $3 \times 3$ identity matrix. Is there a simpler way to solve this problem rather than bashing it out by ...
2
votes
0answers
26 views

number of vertices in a solid

Determine the number of vertices in a solid made up of $x$ triangles, $y$ squares and $z$ pentagons. Without using the Euler's formula $v-e+f=2$ and without counting up all vertices by hand I am not ...