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Questions tagged [geometry]

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

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49 views

Intuition for orientation of tri-vectors in geometric algebra

I am learning geometric algebra from the MacDonald textbook and it states that the outer product is associative. Letting $\bf{u}$, $\bf{v}$, and $\bf{w}$ be vectors $$\bf{u} \wedge \bf{v} \wedge \bf{...
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0answers
12 views

Ratio of axes in an approximate circle

I have some shape that is approximately circular and has area = 100 pixels, with every pixel having area 1. Is there some mathematical way I can define the ratio of the longest axis to the smallest ...
2
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1answer
43 views

Help with homology axioms

I am currently interested in the calculation $H_1(S^2-I)$ where $I$ is an interval embedded in the unit sphere. The answer should be 0, and Hatcher proves it in his book in the sections "Classical ...
6
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2answers
52 views

In an acute triangle ABC, the base BC has the equation $4x – 3y + 3 = 0$. If the coordinates of the orthocentre (H) and circumcentre (P).

In an acute triangle ABC, the base BC has the equation $4x – 3y + 3 = 0$. If the coordinates of the orthocentre (H) and circumcentre (P) of the triangle are $(1, 2)$ and $(2, 3)$ respectively, then ...
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3answers
26 views

What is the value of $a$ where $√a$ is area of a trapezoid which touches the circle with center $O$ (diameter is 2)?

The sides $AB, BC, CD$ of trapezoid $ABCD$ touches the circle with center $O$ and they are equal. $AD$, goes through the point O. If diameter is 2, then the area of the trapezoid is $√a$ . What is ...
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2answers
29 views

Proof for area of an equilateral triangle with respect to one side?

I'm trying to find out the area of an equilateral triangle with respect to one side. Anything wrong with my proof? An equilateral triangle with sides of length $a$ can be divided in half along the ...
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1answer
11 views

spherical polar geometry change in elevation angle

how to calculate change in elevation angle if you know coordinates of two point on surface of sphere. let us say assume that a point move on the surface of sphere from [x1 y1 z1 ] = [0.1 0.1 0.9899] ...
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0answers
26 views

What does this correspond to geometrically?

I recently was playing around with some maths and pondered the following: Let us define $$ \Delta \vec s_{12} = \vec s_1 - \vec s_2 $$ Squaring both sides: $$ |\Delta s_{12} |^2 = |\vec s_1|^2 + |...
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0answers
29 views
+100

What is actually the geometry or analysis behind the fact that $Mob(\hat{\Bbb C})$ is simple?

Let, $Mob(\hat{\Bbb C})$ be the group of all Mobius transformations from the extended complex plane to itself i.e. from $\hat{\Bbb C} \to \hat{\Bbb C}$ . I have been able to prove that (i) $Mob(\hat{...
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2answers
27 views

Find the area of parallelogram and its missing vertex

Given three radius-vectors: $OA(5; 1; 4), OB(6;2;3), OC(4;2;4)$, find the missing vertex $D$ and calculate the area of obtained parallelogram. My attempt: Firstly, we are to find the vectors which ...
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3answers
32 views

Is this a True Property of the Lemniscate of Bernoulli?

I am trying to figure out if the following is true: Take the Lemniscate of Bernoulli (a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus ...
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1answer
33 views

Inscribed circles radius

Let ABCD parallelogram. The inscribed circle in triangle ABD is tangent to BD in E. Show that $$\frac{r_{DEC}}{r_{BEC}}=\tan (\frac{1}{2}\angle ACD)\cdot \tan (\frac{1}{2} \angle ADB)$$ What I have ...
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0answers
19 views

A question about convex bodies

I have been trying to prove that if we have $K\in\mathcal{K}^n$(convex body) centrally symmetric and $H_c=\{x\in\mathcal{R}^n:\langle x,u\rangle=c\}$, where $u\in\mathcal{R}^n$. Then $\text{vol}_{n-1}(...
3
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1answer
43 views

Nine-point circle - proof using plane geometry

I am taking a course in multivariable calculus this year & I thought it would be a good idea to brush up plane and solid geometry. I would like to prove that, for any given triangle, there is a ...
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0answers
35 views

A query on a cyclic pentagon

Let ABCDE be a cyclic pentagon, where AC=2, AD=3, BD=5, BE=1, CD:DE=10:3(Proportion mark, division sign isn't available in my keyboard). What is the value of BC:CE ? I worked with the area of ...
3
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2answers
739 views

sphere-filling curve

Let $S^2$ denote the $2$-dim sphere in $\mathbb R^3$. I am interested in finding a space-filling curve, i.e. a map $\varphi: [0,1]\to S^2$ that is continuous and onto. We know that there is such a ...
2
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3answers
6k views

Solve irregular quadrilateral given 3 sides and 2 angles

I have the following irregular quadrilateral: I know sides T, R, D and angles a and b. How can I determine angles x and y?
2
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1answer
52 views

Proving $ 2 $ angles are equal.

Hi, so I am doing a proof but I need some help proving one part of it. I'm having trouble proving that angle $ D'Bi = $ angle $ D'iB $. point $ i $ is the incenter of triangle ABC and D' is the point ...
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1answer
26 views

Given $n$ points, what is the locus of points $X$ such that the sum of the squares of the distances from $X$ to each point is a given constant?

Given $n$ points $P_{1},P_{2},\dots P_{n}$ and a real number $c,$ find the locus of points $X$ such that $$\sum_{i=1}^{n}XP_{1}^{2}=c.$$ Actually, I'm also interested in a more general case: Given $n$...
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0answers
13 views

What are the general steps to find intersection line between plane and prism?

In the given example it is required to find intersection line between prism and and plane alpha (p,A). Though, I don't understand how we can do that in the given picture. The prism and plane is given ...
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2answers
46 views

What is sufficient to prove Kiselev's Geometry #82?

I am having difficulty realizing what would be sufficient to prove problem #82 asked in Kiselev's Geometry Book I. 82.* On one side of an angle $A$, the segments $AB$ and $AC$ are marked, and on ...
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2answers
21 views

Prove that triangle midline and median split themselves into halves

How can I prove that median and midline in a triangle split themselves into halves? Thanks a lot in advance!
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0answers
18 views

$f^{p,q}:K^{p,q}\to L^{p,q}$ morphism of double complexes($p,q\geq 0$) induces homology iso for fixed $q$. Then $f$ is quasi iso of total complex

Let $K^{p,q},L^{p,q}$ be 2 complexes with $p,q\geq 0$ and $f:K\to L$. Suppose for all fixed $q\geq 0$, $f_\star: H_i(K^{\star,q})\cong H_i(L^{\star,q})$. $\textbf{Q:}$ Do I need $K^{p,\star},L^{p,\...
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1answer
39 views

angles in triangles

In the triangle shown, for $\angle A$ to be the largest angle of the triangle, it must be that $m<x<n$. What is the least possible value of $n-m$, expressed as a common fraction?
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0answers
20 views

Rewrite $ \int_{\mathcal{S}}dP_X=1 $ as conditions on boxes in $\mathbb{R}^d$

Take $r\in \mathbb{N}$ and let $d\equiv r+\binom{r}{2}$. Consider a d-dimensional random vector $X\equiv (X_1,...,X_d)$. Let $P_X$ be the probability distribution of $X$. Assume that $$ \int_{\...
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5answers
410 views

The origin of $\pi$

How was $\pi$ originally found? Was it originally found using the ratio of the circumference to diameter of a circle of was it found using trigonometric functions? I am trying to find a way to find ...
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3answers
3k views

The puzzle of billiard geometry

To hit a bank shot: If the cue ball and the red target ball are the same distance from the rail, then you just aim half-way between them. Referencing the diagram, the angle to aim at is y=ax/(a+b) if ...
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0answers
109 views
+100

If an immersion $X$ maps circles into planes then its image $X(\mathbb{D})$ is homeomorphic to the cylinder.

Let $X:\left( u,v\right)\in \mathbb{D}\backslash \left\{ 0\right\}\subseteq\mathbb{R}^2 \mathbb{% \longmapsto }\left( x\left( u,v\right) ,y\left( u,v\right) ,z\left( u,v\right) \right) \in \mathbb{\...
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1answer
25 views

tilting a disc in 3d space - need help

Lets imagine you have a disc like a CD. Then you take that CD and rest it flat on a desk. Now you tilt the disc left to right and forward to back while touching the desk with 1 point on the edge of ...
2
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1answer
48 views

Topological consequences of negative and zero Einstein condition

Let $(M,g)$ be a complete Riemannian manifold which is Einstein, i.e. $\mathrm{Ric}=kg$ for some constant $k\in \mathbb{R}$. 1) If $k<0$, is $M$ then necessarily noncompact? If so, does the ...
3
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1answer
53 views

In $\triangle ABC$ with $AB=AC$ and $\angle BAC=20^\circ$, $D$ is on $AC$, with $BC=AD$. Find $\angle DBC$. Where's my error?

In $\triangle ABC$ with $AB=AC$ and $\angle BAC=20^\circ$, point $D$ is on $AC$, with $BC=AD$. Find $\angle DBC$. I know the correct solution, but I'm more interested in where is the problem in my ...
0
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1answer
122 views

Rewrite $ \int_{\{(a,b,c)\in \mathbb{R}^3\text{ s.t. } a=b+c\}}dP_{X,Y,Z}=1 $ as conditions on boxes in $\mathbb{R}^3$

Consider a 3-dimensional random vector $(X,Y,Z)$. Let $P_{X,Y,Z}$ be the probability distribution of $(X, Y, Z)$. Assume that $$ \int_{\mathcal{S}}dP_{X,Y,Z}=1 $$ where $\mathcal{S}\equiv \{(a,b,c)\...
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0answers
28 views

Geometric relation

Bonjour and sorry for the bad english. Let ABC a triangle such that $\hat{ABC}=2\hat{ACB}$. The interior bissector of $\hat{ABC}$ intersects the line $(AC)$ at $M$. We put $CM=x, BC=a, AC-AB=d$. ...
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1answer
38 views

Finding the diametre of two circle with the given condition

Two circle as shown in the figure, A is the tangent point of both the circle. B is the centre of the large circle. The distance of CD = 90 mm(according to estimation) and EF = 50 mm. What is the value ...
1
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1answer
21 views

Visualization of 2-dimensional projective transformation

In analytic photogrammetry, there is a transformation called 2-dimensional projective transformation which makes a link between the map plane and the picture plane. If $(X,Y)$ is a point in the map ...
1
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2answers
100 views

Computing the lengths of the obtained trapezium

$ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets $BC$ produced at $P$. My book asked me to show that the area of $APB$ and $ABCD$, are the same, which I did. But it aroused my ...
3
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3answers
244 views

Find out the angle of <ABC

Help me to solve it please.how could it be done.I tried but nothing comes out.Help me please
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2answers
381 views

How do I construct this triangle [on hold]

I was trying to draw the following triangle in latex tikz and I just could not find a way to do it with respect to the given conditions. Is it possible to ...
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3answers
68k views

How to find surface normal of a triangle

If I have a triangle with $3$ points $P_1, P_2,$ and $P_3$, each with $x, y,$ and $z$ coordinates, how do I find the surface normal $N$ in $x, y,$ and $z$ such that $$N_x+N_y+N_z = 1$$ I'm looking ...
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3answers
4k views

How do I find the 3 possible 4th points when given 3 unnamed vertices of a parallelogram?

So, I got have this task: "The points (3, -4, 5), (1, 0, 5) and (3, 1, -2) are three of four vertices of parallelogram ABCD. Explain why there are three possibilities for the location of the fourth ...
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1answer
842 views

Finding the angle of a cone from a 3D point

Given a point in $3$D space $(x,y,z)$ and a circular cone about the $x$ axis, I wish to find the angle of the cone such that the point is on the surface of the cone. For a given point, there is only ...
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0answers
30 views

In triangle $ABD$, there is a cevian from A to segment $BD$. Given $AC =4, CD=12,$ and $AB =8 $,find … [on hold]

In triangle $\triangle ABD$, there is a cevian from $A$ to the side $BD$. Given $AC =4, CD=12,$ and $AB =8,$ find $BC$ if the perimeter of $\triangle{ABC}$ is $30.$ This problem has stumped me for a ...
0
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1answer
36 views

Let ABC be any triangle and let D, E and F be the midpoints of AB, BC and CA. Let X be the point on BC such that AX is perpendicular to BC.

Let ABC be any triangle and let D, E and F be the midpoints of AB, BC and CA. Let X be the point on BC such that AX is perpendicular to BC. Prove that X lies on the circumcircle of DEF. Is it ...
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2answers
69 views

Prove that $PQ=2r$

Suppose that $ABCD$ is a cyclic quadrilateral within a circle of radius $r$. The bisector of the angle $A$ cuts the circle at point $P$ and the bisector of angle $C$ cuts the circle at point $Q$. Then ...
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2answers
59 views

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that only one is a acute angled.

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is a acute angled.
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0answers
12 views

Distance a from point in R3 to a surface defined by a parametric curve and a radius function?

I'm interested in studying the class surfaces defined by: Take an arbitrary parametric curve f : {0..1} -> ℝ3. Pick an arbitrary radius function ...
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1answer
33 views

$N$ points as far apart as possible in a sphere volumetrically?

Given a radius $R$ and points $N$, I want to distribute points in a sphere volumetrically so that they are as far apart as possible. I know that for $N = 1$, I can place it anywhere. For $N = 2$, ...
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1answer
32 views

Find point on curve that has integer coordinates

Given the curve $y=256/x$ find the integer coordinates at which it intersects.
2
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1answer
28 views

For $A_i$ on $y=\sqrt{x}$ and $B_i$ on $x$-axis, with $\triangle B_{i-1}B_iA_i$ equilateral of side $\ell_i$. Find $\ell_1+\cdots+\ell_{300}$.

Let $O$ be the origin, $A_1,A_2,A_3,\ldots$ be distinct points on the curve $y=\sqrt{x}$ and $B_1,B_2,B_3,\cdots$ be points on the positive $X$-axis such that the triangles $OB_1A_1,B_1B_2A_2,...
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0answers
11 views

Estimating the best geocoordinates for two different estimates

I am having two sensors that give me two (latitude, longitude) geocoordinates. For both of them I also know their horizontal accuracy (radius in meters around the geocoordinates). Let's express those ...