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

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arithmetic average over the spherical surface?

intuition behind taking arithmetic average over the spherical surface? . wiki definition :- Consider an open set $U$ in the Euclidean space $R^n$ and a continuous function $u$ defined on $U$ with ...
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Archimedes Classic Proof for Area of Circle: Love it but can't grasp one aspect…

The proof assumes that:... The perimeter of any CIRCUMSCRIBED regular polygon is GREATER than the circumference of the circle. ie: !http://www.themathpage.com/atrig/Trig_IMG/eval1.gif Is this an ...
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1answer
21 views

Length of the side of a regular pentagon is $a$ & length of diagonal is $b$. Value of $\frac{a^2}{b^2}+ \frac{b^2}{a^2}=$? [on hold]

Length of the side of a regular pentagon is $a$ & length of diagonal is $b$. Value of $\frac{a^2}{b^2} + \frac{b^2}{a^2}=$?
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Max value of $9\lambda^2 -2 \mu^2$

Suppose vector $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ such that: \begin{align} \lvert \mathbf{a} \rvert &= \mu \lvert \mathbf{b} \rvert \\ \lvert \mathbf{c} \rvert &= \lambda \lvert ...
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Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter.

Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as diameter. Here is a picture; What I have attempted; Let the ...
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Can a portion of a hypocycloid be a regular polygon?

Hypocycloids are curves that generally don't include straight lines. A significant exception is a hypocycloid with 2 cusps, generated by rolling one circle inside another having twice the radius of ...
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7 views

Monochromatic triangle similar to a given triangle

Given a scalene triangle, $A$ and $B$ play a game. Each move, $A$ chooses a point on the plane, and $B$ colors it red or green. If three points of the same color form a triangle similar to the ...
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Elements of The Excentral triangle

I am learning geometry and found the following question which I partially solved. I want to see the method of approach taken out here in StackExchange. The problem goes as follows: The triangle ...
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2answers
21 views

How do I calculate the height of an arc?

I'm a hobbyist engineer, having one of those moments where my mind goes blank. I know this is a simple problem, but I can't remember how to approach it. I have an arc defined by width and angle. ($w$ ...
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1answer
28 views

Definition of area

I find the question of "What is the area of a circle of radius $5$?" misleading. The answer is $25 \pi$, but what the question really should be saying is "What is the area contained inside a circle of ...
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Area of Elongated hexagon [on hold]

Click to see my picture. Please help me to calculate the actual Area of this one. I can not find the formula to calculate this. Thanks
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A sphere circumscribing a truncated rectangular pyramid

Assume I have a truncated rectangular pyramid with side length $a$, $b$ (large base area) and $c$, $d$ (smaller base area) and height $h$. I would like to find the minimal sphere, which circumscribes ...
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42 views

Prove that $\angle{CBM}=60^{\circ}-\frac{n}{2}$

Given a $\triangle{ABC}$ with $\angle{BAC}=2\cdot \angle{ACB}=n^{\circ}$ where $0<n<120$, let $M$ be an interior point of $\triangle{ABC}$ with $BA=BM$ and $MA=MC$. Prove that ...
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29 views

How to get the properties of an ellipse with six points given.

I am looking for a way to calculate the lengths of both semi-axes and the rotation angle of the ellipse in the image as shown in this picture. Six points are given, with two pairs of points being ...
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13 views

Point halfway around ellipse quadrant

I want to find the length between the centre of an ellipse and a point, P, on the ellipse, where the arc length between P and the intersection of the semi-minor axis with the ellipse is equal to the ...
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transposing a circle onto an angled plane…

good evening. Im no mathematician nor academic and i think what i need is fairly simple, but i dont know how to do it. I have a 6 inch diameter pipe, i need to pass this through a wooden roof at a ...
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32 views

Remaining volume percentage of a sphere

Circular tunnel of radius $r$ is punctured through the center of a homogeneous sphere of radius $R, (r<R)$. What percentage of a sphere is lost? What should be the value of $r$ such that the sphere ...
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33 views

Check if n*m area can be filled with a*b sized tiles and find the remainder [duplicate]

How can I mathematically check if $n\times m$ area can be filled with pieces with $a \times b$ area and find the remainder? For example, can $6\times 6$ area be filled with $1 \times4$ tiles? My made ...
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17 views

Proof: centers of circle tangent to circle and line lies on parabola

first please take a look at this: Given was a circle $c$ with center $A$ and ratio $r$, furthermore three lines $g$, $g1$, $g2$ with: $r = d(g, g1) = d(g, g2)$. Finally, two parabolas $p1$ (and ...
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8 views

What is a metric to determine if a set of points were sampled from a curve?

Suppose we have a curve with its equation given as a spline. We also have a set of ordered $(x,y)$ coordinates. Is there a metric that would indicate whether the points were sampled from the spline? ...
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34 views

Finding the area of the different portions of a rectangle that lie in a grid.

I am an undergraduate student working on a large research project and one part involves calculating the portions of a rectangle that lie in different parts of a cartesian grid. In the figure below, I ...
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4answers
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Unseen Problem based on area of triangle

In $\triangle ABC$, $BD=2CD$ and $AE=ED$, prove that $6\triangle ACE=\triangle ABC$ If $A,X$ is joined such that $X$ is the mid point of $BC$ then: $\triangle ABX=\triangle AXC$ Also, $\triangle ...
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2answers
22 views

Finding the basis of a line on a plane

Say if I have 2 points $P_1$, $P_2$ lying on a plane with known equation $ax+by+cz+d=0$, How do I obtain the expression for the 2 basis (the equations of the 2 red lines; that lie on the same plane) ...
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Transformation of 3D vectors to other planes in 3D

Suppose I have a set of points A, B, C, D, E, F... defined by the 3D vectors AB, AC, AD, AE, AF, AG etc. I can describe the geometry of these by defining them in an arbitrary plane e.g. z = 0 ...
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Given chord length and distance from center find length of a different chord

A chord of length 18 cm is 12 cm from the centre of a circle. How far is a chord of length 10 cm from the centre? I know that chord of equal distance away are equidistant from the centre but these ...
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2answers
29 views

Confusion in Total faces in Cone: 3D

I have checked in many places about how many faces does a Cone have.. As per this link. There is 1 face in Cone As per this link, there are 2 faces in cone As per this Video, A Cone has one face ...
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1answer
24 views

Locus of centres of circles that touch two intersecting circles

let C1, C2 be two circles that intersect each other so we get two Points P1, P2 that lie on both circles. Now we construct a circle $C_i$ with center $M_i$ so that this circle tangents C1 and C2 from ...
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24 views

how to calculate point after rotation

I have basic square with width 36 (red) I'm rotating square by XYZ degrees ... in example there is 45 degrees black square is rotated by point 0,0 green square is rotated by center of sqare 18,18 ...
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1answer
18 views

find the change in coordinate of a point inside rectangle if coordinates of rectangle change

lets consider a point O$(x,y)$ inside a rectangle of having coordinates - a$(x_1,y_1)$, b$(x_2,y_2)$, c$(x_3,y_3)$, d$(x_4,y_4)$. How to calculate the new coordinates of O$(x,y)$ if coordinates of ...
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Writing a linear combination of vectors in a different way

Let $\vec{x}, \vec{y}, \vec{z} \in \mathbb{R}^3$ and let $\vec{z} = \alpha \vec{x} + \beta \vec{y}$ where $\alpha, \beta \in \mathbb{R}$. I want to write an equation involving $\vec{x}, \vec{y},$ and ...
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1answer
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Right triangle inscribed in a rectangle

Please help me on this problem. Points $P$, $Q$, and $R$ are on sides $AB$, $CD$, and $BC$ of rectangle $\square ABCD$ as shown below, such that $|BP|=4$, $|CQ|=18$, $\angle PRQ=90^\circ$, and ...
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2answers
41 views

How to parameterize the interior of a triangle

I would like to know how to parameterize a triangle over $[0,1] \times [0,1]$. I actually only care that the mapping is surjective but a bijection is always nice I suppose. I found this in which an ...
2
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2answers
41 views

Random points on a sphere — expected angular distance

Suppose we randomly select $n>1$ points on a sphere (all independent and uniformly distributed). What is the expected angular distance from a point to its closest neighbor? What is the expected ...
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1answer
44 views

Points of intersection of two circles

We have two circles $x^2 + y^2 = \alpha x$ and $x^2 + y^2 = \beta y $ for any $\alpha, \beta \in \mathbb{R}$. I want to find the points of intersection. We know $(0,0)$ is trivially a solution. If ...
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German translation needed - final sentences of a paper by Hilbert

I am translating a paper by Hilbert into English. I am finished except for the last few sentences, which are confusing me. If anyone can give me a rough/quick translation it would greatly appreciated. ...
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How to derive formula for focus of a parabola?

I understand how to obtain the formula for the vertex of a formula, $ y= a(x-h) + k $ where $ h=-b/2a$ and the vertex is $(h,k)$. However I don't know how to get to $(h,k+1/4a)$. Could someone please ...
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2answers
48 views

Ellipse - relation between a and b such that $F_1P \perp F_2P$

Consider the ellipse $\displaystyle \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$ with foci $F_1 (-e, 0)$ and $F_2 (e, 0)$ (where $e$ is the linear eccentricity). What is the relation between $a$ and $b$ so ...
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1answer
23 views

Prove that for all $n \in \mathbb{N}$ there exists a circle in the $XY$ plane containing $n$ lattice points in its interior.

Prove that for all $n \in \mathbb{N}$ there exists a circle in the $XY$ plane containing $n$ lattice points in its interior. I was trying to use another lemma here that we can always find a point ...
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14 views

Direction of a ray in the hemisphere

If my surface has normal (0,0,1), and I center a hemisphere about that normal, how do I compute the ray that is cast in direction $[\theta, \phi]$ within that hemisphere?
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1answer
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Is it possible to calculate the impact points in this problem?

I am a 18 years old foreign guy with no higher level math education, so I apologize for potentially bad grammar in my question, I hope you will understand what I'm saying nontheless. So I've been ...
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0answers
9 views

Affine transformations in the complex

In $\mathbb C^2$ I have the following three lines: $r_1:3x-y+3=0, r_2:y=0, r_3:x-i=0$ I want to find all the affine transformations such that $f(r_1)=r_2, f(r_2)=r_3, f(r_3)=r_1$ How can I do it? ...
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1answer
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How to find the slope of a line when you only have a point and an angle?

A line passing through $(4,7)$ makes an angle of $45$ degrees with the $y$-axis. How do I find the slope of this line?
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2answers
58 views

Do there exist geometric figures with multiple centers of symmetry?

Do there exist geometric figures with multiple centers of symmetry? Under what conditions does the center symmetry lie inside a geometric figure? Recall the definition of a center of symmetry: A ...
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1answer
53 views

How can you divide an octagon into 5 equal parts?

How would you divide an octagon into 5 equal parts? This is a question that we are working on in 2nd grade. Do you have an answer for us? Thanks, Mrs. Parsons Class West View Elementary Burlington, ...
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2answers
21 views

Three vertices of a Paralellagram

Three vertices of a parallelogram have coordinates (0,1), (3,7), and (4,4). Name a point that could represent the fourth vertex of the parallelogram..
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1answer
27 views

Geometric intuition of the equation of a plane

Let $\pi$ be a plane in an $d$-dimensional space with normal vector $\underline{w} = [w_1, \dots,w_d]^T$. If point $\underline{p} = [p_1, \dots,p_d]^T$ is in the plane and $\underline{x}= = [x_1, ...
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0answers
9 views

maximal volume/diameter of a set of simplexes

I am trying to develop a simplicial integral in $R^n$ and I am faced with the problem of controlling the "compacity" of a set of simplexes: Let $S$ be a finite set of n-d simplexes in $R^n$. Define ...
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1answer
32 views

Rotation about z axis using quaternions

I am working with quaternions and rotation, but I am missing something about how a rotation expressed as a quaternion works. I also discovered that there are different convention for quaternions (JPL, ...
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2answers
53 views

Can $n$ circles be drawn such that all have a common intersection but no two intersect individually

I was fiddling with plane geometry when a question came into my mind: Can $n$ circles ($n \ge 3$, $n \in \mathbb{N}$) be drawn such that: $C_1 \cap C_2 \cap C_3 \cap \ldots \cap C_n \not = ...
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2answers
45 views

Triangle with a bisected side and a trisected side

In the figure, $P$ is the midpoint of $BC$, and $AR=RQ=QC$. Prove that: $BR=4SR$ $\triangle ABC =12\triangle ASR$ From the given figure, 1). $BP=PC$ 2). $\triangle APB=\triangle ...