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
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Maximising the Area of a Cyclic Quadrilateral

In cyclic quadrilateral $ABCD$, $AB = AD$. If $AC = 6$ and $AB/BD = 3/5$, find the maximum possible value of $[ABCD]$. (Source: SMT 2014) If we let $AB=AD = 3x$ and $BD=5x$, from Ptolemy, we have ...
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2answers
37 views

Circle with center point and tangential to lines

I have defined Points all points (3 blue, and one green). All points have the same distance to A point. Yellow lines are bisectors. I have equations of AB and ...
2
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0answers
11 views

How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere of radius $R$?

Question How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere $\mathbb S $ of radius $R \quad (R \gg a)$? What I have thought so far Since the area of the ...
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2answers
21 views

Find point on line withv given start point, distance, and line equation

I have line equation $$ Ax +By + C = 0.$$ I have start point (on this line): $ P_0 = (X_0, Y_0)$. I have distance $d$ too. I need find point $P_2$ with distance $d$ from $P_0$ and placed on ...
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1answer
29 views

Euclidean metric formula

Is this the correct formula for the euclidean metric (in $R^4$ )? $g_E = dr^2 + r^2(d\theta^2 + d\phi^2 + d\tau^2 + \cos \theta d\tau d\phi).$ I have been doing some calculations that are wrong and ...
0
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1answer
27 views

Is there a formula for finding the centers of the faces of a platonic solid?

Is there a formula for finding the centers of the faces of a platonic solid given the center of the first (origin) face to be $P_0(x_0,y_0,z_0)$?
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0answers
36 views

Area of a circumcenter triangle equals area of medial triangle

Let $X$, $Y$, $Z$ be the midpoints of sides $BC$, $AC$, $AB$ respectively in triangle $ABC$. Let $O_{A}$, $O_{B}$, and $O_{C}$ be the circumcenters of triangles $AZX$, $BXY$, and $CYZ$ respectively. ...
4
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1answer
45 views

Geometry construction problem

Given two circles $S_1$ and $S_2$, a line $l_1$, and a length $a$ that is less than the sum of the diameters of the circles, construct a line $l$, parallel to $l_1$, so that the sum of the chords that ...
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2answers
78 views

What is the geometric interpretation of a vector squared?

I'm working through Introduction to Space Dynamics by William Tyrrell Thomson. I am having to do a lot of research to make it through even small parts, but I am unable to find information to make me ...
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2answers
16 views

different formulae to find aspect ratio

I am working on a software 2D model where we need to work with aspect ratios. My boss gave us a formula which defines it as: ...
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1answer
37 views

Easy question about line up points ?

Well, there is a demonstration to prove that 1+1=2 so is there a demonstration to prove if you take two points on the space, they are line up ? Thank's
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0answers
36 views

Geometric proof of BD/BE = CD/CE ; methods for congruent triangles

It seems to me that if BAC = DEA this would be straightforward, but I lost myself in the variety of congruencies here.
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0answers
42 views

General form for the rotation of a function.

When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely. This was what I did to rotate $y=2x+1$ by ...
2
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0answers
54 views

Which convex $2n$-gons have symmetry group $D_n$ instead of $D_{2n}$?

The equilateral octagon $M$ in the first image has the same symmetry group as the small embedded square - namely the dihedral group $D_4$ - with $8$ elements and generators ${x,y}$ with $x^4 = e, y^2 ...
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3answers
56 views

Distance between the nail and the center of the disk

Suppose you have a disk with radius $r$ and a string of length $2 \pi r+l$, i.e. longer than the perimeter of the disk. Hang the disk (of center $O$) from the nail at $A$ using the string as shown ...
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1answer
22 views

Geometry - Volume of a distorted tent

How would one calculate the volume of a tent shaped object with the upper edge not parallel with the base plane of the tent? edit: The tent has a rectangular base with two poles at different heights ...
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1answer
30 views

How can you prove that perimeter of right triangle equal to diameter of incircle and twice the diameter of circumcircle? [on hold]

How can you prove that perimeter of right triangle equal to diameter of incircle and twice the diameter of circumcircle?
2
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1answer
25 views

Are the principal congruence subgroups of SL(2,Z) normally generated by a single element?

Let $N\ge 3$, then would I be correct in saying that the principal congruence subgroup $\Gamma(N)$ (defined to be the 2x2 matrices in $SL(2,\mathbb{Z}$) congruent to the identity mod $N$) is the ...
3
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3answers
416 views

How did the Ancient Greeks know that the circle method of finding square roots was mathematically valid? How do we know that?

The Ancients used this method. (or at least James Grime said in a numberphile video) To construct the square root of a number, draw an interval of length $a+1$, and then draw a semi-circle with the ...
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2answers
31 views

Space formed by dot products of three vectors

Suppose I have 3 3D unit vectors $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{V}$. I define the three corresponding scalars $u_1=\mathbf{v}_1 \cdot \mathbf{V}$, $u_2=\mathbf{v}_2 \cdot \mathbf{V}$, ...
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3answers
58 views

How to solve for $\theta$ in an expression involving linear and $\sin$ terms

While trying to solve a spatial geometry problem, I came across the expression: $$156θ-36\sin\theta=554.8$$ And I have no idea where to even begin.
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4answers
39 views

Ellipse focal proof

In the ellipse with equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ How to proof the production of distances between focal and a random tangent is $b^2$ $F1A * F2B = b^2$
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0answers
28 views

Partial Integral of an ellipse

this is my first question on stack exchange so please bear with me. I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel ...
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0answers
25 views

Isoceles Triangles on a Grid Proof

Given: A Finite Set of Unit Squares on a Large Grid. If we were to choose one of those sets of unit squares, we see that the squares of the set are tiled with isoceles right triangles, each with a ...
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0answers
20 views

Convex basis and conical basis (how to draw?)

There is a question, I'm struggling with: Find vertices of the following described polyhedron, $P:=P(A,b)=conv(V)+cone(E)$ where $V$ is the set of all vertices of $P$ and $E$ is the set of all ...
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1answer
15 views

Calculating my location based on known location

This question is linked to Can known object be used to back-calculate my location? (been almost a month, figured it would be best to start a new question.) I have a map, and I know which way true ...
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0answers
50 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
2
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1answer
26 views

Does there exist a 4D torus with a spherical cross-section, analogous to a circle for the 3D case?

I don't mean to be a bother. It seems as though the answer may be obvious, but then, seemingly simple math questions can have surprising answers. I should also like any pointers re: the general ...
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0answers
33 views

Show that the Axioms are satisfied [on hold]

A real number x is called dyadic, if it can be written as $x = a/2^n$, for some integers $a,n \in \Bbb Z$. In particular, all the dyadic numbers are rational numbers. Let $\Pi = \{(a/2^m, b/2^n)|a, b, ...
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1answer
21 views

Polygonal sides when interior angle relationship is given

The difference between any two consecutive interior angles of a polygon is 5°. The smallest angle is 120°. Find the number of sides. I know that the sum of interior angles of a polygon is ...
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1answer
71 views

Can anyone tell me how to factor this expression? [on hold]

...$\dfrac{2x^2-4x}{x+10}$ this is in response to a side splitting theorem question where these values are part of the proportions. Here is the complete question: We have 2 similar right triangles. ...
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1answer
32 views

Proof involving circumradius of triangle and Law of Sines

Show that in any triangle, we have $ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right), $ where $R$ is the circumradius of the triangle. I'm not quite ...
2
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1answer
19 views

What is the solid angle of the intersection loop between a cone and an off-axis sphere?

An upright (green) cone with opening angle $2a < \pi/10$ has its vertex at point O with cartesian xyz coordinates $(0,0,0)$. The cone axis (dotted line) lies in the plane $y=0$ and is parallel to ...
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1answer
26 views

Show that the midpoint of $AB$, $AC$, and $DE$ are aligned.

Let $ABC$ be a rod, $D$ and $E$ two points such as: $\vec{EC} = k \cdot \vec{EA} / \vec{DA} = k \cdot\vec{DB}$. How can I show that the midpoint of $AB$, $AC$, and $DE$ are aligned?
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1answer
15 views

isoperimetric inequalities in permutohedron

Consider the graph whose vertices are all n! permutations of numbers 1..n and there is an edge between two vertices iff we can get from one to another by an adjacent transposition. We call this graph ...
3
votes
1answer
48 views

Rectangle circumscribed to an ellipse of max area/perimeter

I could solve the classical problem of maximizing the area (fixing the perimeter) or maximizing the perimeter (fixing the area) of an inscribed rectangle, but I don't know how to solve ...
2
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2answers
43 views

Jacobi Elliptic Functions Special Case

I have spent some time analysing the pendulum problem, and hence the Jacobi elliptic functions recently, and have come across what seems to me to be a slight inconsitency. I define my $am(t|k)$ as the ...
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0answers
21 views

hollow sphere problem [on hold]

A hollow space on earth surface is to be filled. Total cost of filling is 20000 Rupees. The cost of filling per mt3 is 225 Ruppes. How many times a size of 3 mt3 soil is required to fill the hollow ...
0
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1answer
30 views

Fattest scalene quadrilateral

What angles of a plane scalene quadrilateral maximize its area? By 'scalene' I mean the four lengths are unequal. It is known that if a quadrilateral has opposite sides equal and parallel as a ...
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0answers
26 views

Geometrical interpretation of an overidentified linear system

In my econometrics class we talked about Instrumental Variables. Suppose one has a $n\times k$ matrix $X$ of regressors and a $n\times m$ matrix $Z$ of instrumental variables. Given the matrices are ...
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1answer
32 views

Find part of segment between two circle centers

I drew the following image to help me explaining the question: Having two circles Source and Target, I want to build an arrow like in the image. The Source has coordinates $Source(sx, sy)$ and ...
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0answers
44 views

An Interesting Areas Question [on hold]

Let the area of the triangle $ABC$ be $x$. The points $A_1$, $B_1$ and $C_1$ are the mid points of the sides $BC$, $CA$, and $AB$ respectively. The point $A_2$ is the mid point of $CA_1$. Lines ...
2
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1answer
79 views

Surface normal to point on displaced sphere

I want to calculate the surface normal to a point on a deformed sphere. The surface of the sphere is displaced along its (original) normals by a function $f(\vec x)$. In mathematical terms: Let ...
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1answer
33 views

Ratio of squares touching edge of circle?

Consider an infinite amount of squares stacked on top of each other where the top left corners are touching the edge of a circle: Call the largest blue square x. How would I find the ratio of ...
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2answers
30 views

Construct a midpoint of two parallel lines with only straightedge

Say I have a large plank of wood that I'm trying to cut in half the long way, but I only have a straightedge (no compass). How can I mark the midpoint between the long edges? (As a note, this isn't a ...
2
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1answer
44 views

Finding the radius of a third tangent circle

Sorry if this is a foolish question, but I'm having difficulty understanding how to solve for $r_3$ in the following diagram... According to WolframAlpha's page on tangent circles, the radius of ...
3
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1answer
42 views

Inequality between area and boundary length, $4\pi A \leq L^2 $

Suppose we have a simply connected region $R$ in $\mathbb{R}^2$ with area $A$ and the boundary of $R$ is a curve sufficiently well behaved (say piecewise $C^1$) that we can say it has length $L$. Then ...
0
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1answer
20 views

The vertical projection of a chord of a circle?

I was wondering if anyone could help me with the problem below (finding x): So we are given t_i (the initial tangent angle to the circle), t_o (the exiting angle of the tangent of the circle), the ...
2
votes
1answer
46 views

Creating an ellipsoidal 3D surface

I am trying to find the equation of a 3D ellipsoidal surface. I have thought of two approaches which are schematically shown below: By revolving an elliptical arc over a 3D elliptical path: Or by ...
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0answers
20 views

Projection of a Triangle into a Tetrahedron

I was referring to a paper to implement an algorithm in which one of the step was to project the triangle into the ...