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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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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 ...
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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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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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1answer
44 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 ...
2
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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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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}(...
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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
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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1answer
34 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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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 ...
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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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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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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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0answers
22 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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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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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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2answers
47 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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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 ...
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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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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 ...
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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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2answers
383 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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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 ...
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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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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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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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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 ...
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0answers
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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 ...
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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
32 views

Find point on curve that has integer coordinates

Given the curve $y=256/x$ find the integer coordinates at which it intersects.
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2answers
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Get leg length depending on another leg knowing the perimeter of a rectangle triangle [on hold]

Knowing the perimeter of the triangle, how can i find the side (leg) b in function of the leg a and it's perimeter p.
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2answers
48 views

In acute $\triangle ABC$, show $DE+DF \leq BC$, where $D$, $E$, $F$ are the feet of the altitudes from $A$, $B$, $C$, respectively.

Let $\triangle ABC$ be an acute angled triangle. The feet of the altitudes from $A$, $B$, and $C$ are $D$, $E$, and $F$, respectively. Prove that $$DE+DF \leq BC$$ and determine the triangles ...
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Why don't circles have infinite degrees? [duplicate]

A triangle has 180 degrees. A square has 360. A pentagon has 540. A hexagon has 720. An octagon (which is starting to look a lot like a circle) has 1080. You see the trend?
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0answers
8 views

Orbit of a point and symmetries of a specific graph

Given is the graph: I am interested in determining the orbit of the point 1 and also to determine the amount of symmetries that fix each of the points 1, 2 and 3? My approach: Notice that the ...
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0answers
54 views

How do you determine if a geometric construction has degrees of freedom?

I would like to know if there is a common approach to proving or disproving whether degrees of freedom exist after following a geometric construction scheme? To clarify, that the result of the ...
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1answer
76 views

Side of the equilateral triangle

I tried very much but since tomorrow is my exam, i cannot risk it. The following is a geometry problem, which i have tried very much but could not grasp a solution. I think that i require pythagoras'...
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1answer
12 views

Sphere inscribed in a cone

If a cone of height h and radius r has a sphere inscribed in it such that it touches the base and the curved surface area, how can I find the radius of the sphere? (Is this in the level of a 9 grader?)...
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1answer
29 views

How to layer objects - geometry?

I'm developing a kind of perspective based 2d/3d game. I've got an X- and an Y-axis like I've displayed in the image below. To my question: I've got a bunch of objects (marked with "1" and "2") on ...
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3answers
25 views

Prove that every tangent of a function cuts $y$ axis at a point that is at equal distance from (0,0) and touching point

If a function y = $\frac{1}{2}$$\sqrt{x-4x^2}$ is given, how would one prove that every tangent of the function cuts $y$ axis in a point that is at equal distance from point $(0, 0)$ and the point at ...
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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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2answers
39 views

Three equilateral triangles form a hexagon [on hold]

As I posted yesterday, I was learning about vectors yesterday. I know how to add and subtract them, but I can’t multiply yet. So here is an extra problem from my teacher I need help with: Given a ...
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 ...
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0answers
59 views

Find a point $X$, in the plane of regular pentagon $ABCDE$, that minimizes $\frac{XA+XB}{XC+XD+XE}$.

Find such a point $X$, in the plane of the regular pentagon $ABCDE$, that the value of expression $$\frac{XA+XB}{XC+XD+XE}$$ is the lowest. I tried using Ptolemy's theorem but don't know how to make ...
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1answer
37 views

Let $H$ be the orthocentre of triangle $ABC$. Prove that the midpoints of $AB, AC, AH, BC, BH$ and $CH$ form a cyclic hexagon.

Let $H$ be the orthocentre of triangle $ABC$ ($H$ is the point inside triangle $ABC$ such that $AH ⊥ BC$, and $BH ⊥ AC$ and $CH ⊥ AB$). Prove that the midpoints of $AB, AC, AH, BC, BH$ and $CH$ form ...
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0answers
31 views

how to obtain peripheral recangle of arbitrary ellipse?

Suppose have arbitrary ellipse with center $(x,y)$ and its radius $(a,b)$. I want obtain rectangle that sides tangent of peripheral ellipse. the below image describe issue :
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2answers
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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
68 views

What is the area of this figure?

I want to know how to find the area of this shape: Yellow, white and blue shapes are ellipses. Red is a square. The blue ellipse is not cut in half by the square. I know that I have to add up all ...
2
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1answer
44 views

Hexagon not regular

Plotting the set: $$ \small \left\{ (x,\,y) \in \mathbb{R}^2 : x + \frac{3}{2} \le y \le x + 2, \; 3\,x \le y \le 3\,x + 1, \; -2\,x + 3 \le y \le -2\,x + 4 \right\} $$ the following hexagon is ...