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

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Discovering length of line

I'm attempting to work out length of BD from below diagram : The length of BD is -2 +- some value. But since I do not know the y co-ordinate of B can the length of BD be determined from ...
2
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0answers
31 views

Derive equation for shear modulus $G=E/(1+2v)$

shear modulus, G young's modulus, E and Poisson's ratio, $v$: $G=E/(1+2v)$ I have always wondered how this relation is derived, but have never found a derivation that I could follow online. I ...
2
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5answers
76 views

How to find the coordinates where the altitude of a triangle intersects the base in 3 dimensions?

Assuming I know three completely random coordinates in 3d space that correspond with vertices of a triangle, how can I then find the point at which the altitude intersects the base? I know how to ...
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1answer
33 views

Triangle Inequality Problem…

In triangle $ABC$, the medians $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ concur at the centroid $G$. (a) Prove that $AD < (AB + AC)/2$. (b) Let $P=AB+AC+BC$ be the perimeter of $\...
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1answer
27 views

Calculating relative position of points when zoomed in and enlarged by a rectangle

There is a rectangle, defined by the top left point $R1(0, 0)$ and the bottom right point $R2(200, 200)$ (the $y$ $axis$ is inverted). In that rectangle, there are some points $P1(100, 100)$, $P2(50, ...
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1answer
30 views

Length and diameter of a spiral of nanotube

I was reading from this popular article (in french). Talking about nanotubes of carbon the author says (my translation): The diameter of the nanotube is of the order of a millionth of a millimiter....
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1answer
38 views

A Question about Triangle Inequality [duplicate]

Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side, if it is a positive integer? I'm not sure if the word "acute" affects the problem. If not, is the ...
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2answers
36 views

Find the area of $S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$

Let $S$ be the domain defined by $$S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$$ find the area of $S$ This is middle school problem,so I think it ...
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2answers
36 views

Ellipse equation from center and point on ellipse [closed]

Is there a way to get the equation of an ellipse iw we know the center and one point on the ellipse ?
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3answers
50 views

How many combinations are there for the interior angles of a triangle?

Suppose the interior angles of a triangle are all Natural numbers. How many combinations of angles are there without repeating similar triangles? So for instance, {1,1,178}, {1,2,177},...But without ...
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0answers
25 views

How to visualize this geometric setup?

I am working on a derivation (physics) and the first part requires one to imagine a geometric setup. Since I'm not a native speaker of the language, I am having trouble in doing so. The relevant ...
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1answer
35 views

Why should sum of coefficients of collinear position vectors be 0?

Suppose $a,b,c$ are collinear position vectors then we know that $xa+yb+zc=0$ where $x,y,z$ are scalars. But in my book something additional is also mentioned that $x+y+z=0$,why must it be so ? I ...
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1answer
46 views

How to find the area shared by 4 quadrants inside a square?

I was to find the blue area in this question : As described about how it's a square with 4 quadrants of same radius intertwined with each other, now to find the blue part area I thought about ...
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0answers
46 views

Cutting a pie into 2 unequal peices with a single cut, minimising its length. [closed]

Suppose we have a circle with an area of 1, which we are to cut into two pieces, of area (x) and (1-x) respectively. Let x<0.5. How should we make the cut, to minimise its length? What is the ...
2
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1answer
29 views

Does shearing a sphere generate an ellipsoid?

In my preferred 3D modeling software I see that however I shear a sphere, I seem to be able to make a nearly identical shape using some combination of non-uniform scaling followed by rotation. Are ...
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0answers
53 views

What is the mathematical vernacular for $\Delta x$

I want to use the proper terminology when I discuss length scales associated with $\Delta x$, where $\Delta$ is the difference operator. In other words $\Delta x = |x_1-x_2|$. It is a measure of the ...
2
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1answer
52 views

Geometry Triangle Question 3

In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there? Two sides of an acute triangle are 8 and 15. How many ...
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1answer
23 views

Change of coordinates in $R^{n}$ where the diagonal goes to $x=0$

Say I have a system of coordinates $\{y_{1},y_{2},...,y_{n}\}$. I'd like to get a new system of coordinates $\{x_{1},x_{2},...,x_{n}\}$ where the diagonal $y_{1}=y_{2}=...=y_{n}$ is $x_{1}$, and I'd ...
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2answers
50 views

Geometry Questions: Triangles [closed]

Can you guys please help me with these problems? In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there? The ...
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0answers
12 views

Looking for a particular parameterization of $S^n$

Say we have take vectors $(x_1,..,x_d) \in S^{d-1}$ and we look at vectors $(a_1,..,a_d) \in (\mathbb{Z^+ \cup \{0\}})^d$ such that $\sum_{i=1}^da_i =k$ for some positive integer $k$. Is there any ...
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0answers
34 views

Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
2
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0answers
44 views

Find the ratio of slope

Note : Elevation $46000$ and all dimention in $mm$ (milimeter) The pipe will be installed on a surface of module structure, that module structure has different surface. I want to know " ratio ...
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1answer
30 views

Question re: derivation of formula for volume of cone

The diagram above comes from a derivation of the formula for the volume of a cone; it's one of the preliminary steps, and sadly, I'm stuck on it. What we're doing here is inscribing an "infinite" ...
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1answer
33 views

Linear application

Let $f:\mathbb{R}^3\to\mathbb{R}^3$ be a linear application and let $\{e_1,e_2,e_3\}$ the canonical basis of $\mathbb{R}^3$. We know that $\operatorname{Im} f=\langle(1,1,3), (0,1,1)\rangle$ and that ...
3
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1answer
35 views

Area of circle segment intercepted by a line

The problem I want to solve is to calculate the filled area in the following diagram - so basically the area between the two circular arcs but with the red line cutting off one side. I think I have a ...
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1answer
60 views

Finding centers of ellipses with two points and their respective tangents

I hope you can help me with the following, probably rather complex dilemma: I generally want to find an ellipse given two points and their respective tangents in 2-D space (X and Y coordinates). Now ...
2
votes
3answers
57 views

Sphere and tetrahedron

If we have sphere inscribed in a tetrahedron, and if the distances from the center of the sphere to the edges of the tetrahedron are equal, is it true that this tetrahedron is always regular? I'm ...
4
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0answers
78 views

How did Archimedes calculate rational bounds of pi from a 96-gon?

Archimedes famously determined that $223/71 < \pi < 22/7$ using the 96-gons circumscribed by and circumscribing a circle of unit diameter. But I haven't found a reference that explains the final ...
0
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1answer
22 views

A straight line moves so as to meet the straight lines

A straight line moves so as to meet the straight lines $y=mx, z=c$ and $y=-mx, z=-c$ in A and B and intersects the curve $yz=k^2, x=0$, show that the locus of the middle point of $AB$ is $$(m^...
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0answers
12 views

What is the set of 4 coterminal symmetric rays called? [duplicate]

The minimum number of coterminal symmetrical rays is four. I would like to pursue this unique feature further. I do not find the names Star, Caltrop, satisfactory for this structure. I would like to ...
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2answers
65 views

Geometric interpretation of a complex set

These usually aren't too bad but I had difficulties thinking of what the set $$\{z\in\mathbb{C}:|z+i|=2|z|\}$$ looks like in the complex plane. I got as far as $$|z+i|=2|z|\Rightarrow \sqrt{(z+i)(\...
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1answer
24 views

Find the volume of a rectangular parallelpiped

Can anyone help me with this problem? I used the following drawing in solving this problem and got a wrong answer. I don't know whether it is due to my misunderstanding of the problem. Here is my ...
0
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0answers
28 views

Lines tangent to two circles

I'm trying to find the lines tangent to two circles. I've seen several examples but with poorlyy explained methods. Given the circle $(x-x_{0})^2+(y-y_{0})^2=r_{1}^2$ and the the line equation $y=...
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0answers
47 views

Prove that the following statements are equivalent.

Let $w_1,w_2,\gamma,\delta\in\Bbb R^n$. We define $t_{\gamma}=\frac{1}{1-\frac{w_2.\gamma}{w_1.\gamma}}$ and like this is $t_{\delta}$. Prove that the following statements are equivalent: $t_{\gamma}...
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1answer
39 views

Prove concurrency in a triangle

If a circumference cuts a triangle $ABC$ at its sides $BC$, $CA$ and $AB$ at points $P, P'; Q, Q'; R, R'$; respectively (so twice on each side, and if $AP, BQ$ and $CR$ are concurrent (intersect at a ...
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0answers
21 views

when can i move a sum through this tensor product?

If I have a vector space $V^{(1)}\otimes V^{(2)}$ and I have some ray $\sum\limits_k x_k s_k\otimes s'_k = s\otimes \sum\limits_k x_k s'_k$, is the only solution that $s_k = s$ $\forall$ $k$? All $x_k$...
3
votes
3answers
80 views

Angle between 3 points

I have three points $(x_1, y_1), (x_c, y_c), (x_3, y_3)$, where I know $(x_1, y_1), (x_c, y_c)$, the angle $\theta$, and $c$ on the dash line in the following figure. How to calculate the point $(x_3, ...
2
votes
3answers
172 views

Prove triangle similiarity by given expression

I am working on the following problem, but I can't seem to figure it out. The length of the sides in the triangle $T_1$ are $a_1$, $b_1$ and $c_1$. The length of the sides in the triangle $T_2$ ...
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1answer
79 views

Geometry-Triangle

Let $ABC$ be a triangle with $DAE$, a straight line parallel to BC such that $DA=AE$. If $CD$ meets $AB$ at X and $BE$ meets $AC$ at $Y$, prove that $XY$ is parallel to $BC$ I tried to use the angle ...
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1answer
28 views

Switching Cartesian and Polar Coordinates

I passed by this image long time ago, and I am now wondering if this can happen anytime. Suppose we have the cycloid, given with equations $x=R(t-\sin{t}), y=R(1-\cos{t})$. On wikipedia, it says that ...
0
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2answers
51 views

Angle Between Two Tangents

In the picture below, the angle $AOB$ is $\delta \theta$, and then it is deduced that the angle between the two tangents is the same from the fact that the angles in a quadrilateral add up to $2 \pi$. ...
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2answers
46 views

How can I imagine double/repeated root of a quadratic equation?

A quadratic equation such as $(x-2)^2=0$ has a repeated root $(2,2)$. A lot of things in math (equations, matrixes and so) can be nicely drawn on a $2D, 3D$ etc plane (with $x$-$y$ axis). I mean, I ...
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1answer
34 views

Question from triangles [closed]

in 🔺ABC, P and Q are points on sides AB and AC respectively, such that PQ||BC . If AP=2.4 cm, AQ=2cm , QC=3 cm and BC=6cm , find AB and PQ?
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1answer
90 views

Why is the dual cone of $l^1$ is $l^\infty$?

I just noticed somewhere in Convex Optimization that the dual cone of $l^1$ is $l^\infty$! (A diamond in $\mathbb{R}^2$ for $l^1$ is a square in $\mathbb{R}^2$ for $l^\infty$.) In fact I cannot ...
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1answer
21 views

lines of bearing bounding a circle

diagram Given a circle of radius R, that is D units from the origin, I would like to find a method for finding the two lines of bearing from the origin that are tangent to to the circle. This is not ...
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1answer
29 views

Formula for 2d coordinates of a Steiner Point of a triangle

I've searched around a bunch, and I still haven't managed to find any clear-cut way to find the x and y coordinates of the Steiner point defined by the coordinates of exactly 3 points. Does anyone ...
1
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1answer
21 views

Geometry, triangle incenter problem

I is the incenter of triangle $ABC$. $X$ and $Y$ are the feet of the perpendiculars from $A$ to $BI$ and $CI$. Prove that $XY$ is parallel to $BC$ I tried to use the angles $AXI$ and $AYI$ to prove ...
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1answer
48 views

Algorithm for getting consecutive line segment edge points from midpoints

So I have a rectilinear grid that can be described with 2 vectors. 1 for the x-coordinates of the cell centres and one for the y-coordinates. These are just points with spacing like x spacing is 50 ...
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1answer
38 views

Working out length of side of triangle?

I'm taking mooculus course from https://mooculus.osu.edu/exercises/linearTriangles1 and am given following problem : What is the intuition of the hint : 'length of DA = abscissa of D minus abscissa ...
0
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1answer
35 views

Two chords $AB$ and $CD$ intersects

Two chords $AB$ and $CD$ intersects at point $P$ at right angle. $M$ is mid point of $BD$. $MP$ is produced to $N$ on $AC$. Prove that $PN\perp AC$ My Attempt $\angle APN=\angle MPB$ $\angle BPD=...