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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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0answers
60 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
30 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
27 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 ...
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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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 ...
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1answer
55 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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1answer
102 views
+50

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
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
69 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 ...
2
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1answer
25 views

Two regular pentagons and the sum of vectors connecting their vertices

Today I was learning about vectors. The teacher gave me the following problem: Consider two regular pentagons $A_1A_2A_3A_4A_5, B_1B_2B_3B_4B_5$ on the plane. The center of the first pentagon is $O_A$...
2
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1answer
27 views

Find the atractor of the triangles formed by joining the feet of altitudes of the previous triangle?

Triangle 1 (see the picture) is given. Find the point toward which the vertices of triangle n -> infinity converge, assuming that triangle n is constructed by uniting the feet of the altitudes of ...
5
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4answers
75 views

Given a chord, how do I find the ellipse?

It will explain my use case at the end, in case I am approaching this wrong, but I will start with the math question. Given: a point $\rm P$ on an ellipse; the slope of the tangent (or normal) to ...
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0answers
37 views

A rectangle ABCD is given. n different points are chosen between A and B. Similarly, m different points are chosen on the side AD.

A rectangle ABCD (5 units x 4 units) is given. On the side AB, n different points are chosen strictly between A and B. Similarly, m different points are chosen on the side AD between A and D. Lines ...
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1answer
39 views

Two solid cylinders are mathematically similar. The sum of their heights is 1. The sum of their surface areas is 8$\pi$.

Two solid cylinders are mathematically similar. The sum of their heights is 1. The sum of their surface areas is 8$\pi$. The sum of their volumes is 2$\pi$. Find all possibilities for the ...
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1answer
47 views

Let S be a semicircle with diameter AB. The point C lies on the diameter AB and points E and D lie on the arc BA, with E between B and D.

Let $S$ be a semicircle with diameter $AB$. The point $C$ lies on the diameter $AB$ and points $E$ and $D$ lie on the arc $BA$, with $E$ between $B$ and $D$. Let the tangents to $S$ at $D$ and $E$ ...
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 ...
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1answer
26 views

Worst-Case Addition to Smallest Enclosing Circle

Imagine you have a convex polygon $p_1$ and put a smallest enclosing circle $SEC_1$ around it, the lengths of each side of the polygon $l_i$ can be anything and don't have to equal each other. Now, ...
1
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1answer
59 views

Circle rotating in circle rotating in circle

There are three circles with different radii arranged such that the smallest circle is contained entirely within the next larger circle and that circle is entirely contained within the largest circle. ...
0
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1answer
23 views

Angle between sum of vectors

Let $u,v$ and $w$ be vectors in $\mathbb{R}^n$ and let $\theta(u,w), \theta(v,w)$ and $\theta(u+v,w)$ represent the angle between each listed pair of vectors. Does it hold that one of the following ...
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1answer
33 views

Determining the value of AH\AD+BH\BE+CH\CF where H is orthocentre of the three diagonals AD, BE and CF in an acute-angled triangle ABC.

In an acute-angled triangle ABC, AD, BE and CF are respectively perpendicular to the opposite side of the three climax point included A, B and C. H is the orthocentre of the orthogonals. What is the ...
5
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4answers
1k views

Having a cube, with a point at its center. What are the points that are equidistant from the center point to the cubes vertices?

Having a cube, with a point at its center. What shape do the points wich are equidistant between the center and the cubes vertices make? The source of why I had this question is the following photo ...
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1answer
37 views

Area of a circle segment on sphere, given radius (meters) and central angle (degrees)

Situation I have a circle segment and some information about the circle it belongs to. Given Information: radius of the circle in meters central angle in degrees lat/long of all three points on the ...
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3answers
50 views

Are all matrices almost diagonalizable?

Every real $2$ by $2$ matrix that is not diagonalizable is similar to the $2$ by $2$ jordan canonical form, $$ J_2=\begin{bmatrix}s&1\\0&s\end{bmatrix}, $$ where $s$ is the eigenvalue (with ...
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2answers
27 views

The Domain of a continuous function is an open set

I have started studying the book "Differentiable manifolds; An Introduction" by Brickell & Clark. But I have encountered the following paragraph on page 4. Why is the domain of a continuous ...
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2answers
36 views

Find the locus of the middle point of the intercept on the line y=x+c made by the lines 2x+3y=5 & 2x+3y=8, c being a parameter?

Here's my shot: since the two lines are parallel, I figured that the middle point should be equidistant from the parallel lines,so using distance formula:- $ \frac{2x+3y-5}{\sqrt{13}}=\frac{2x+3y-8}{\...
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0answers
43 views

Deriving the general equation of ellipses in cartesian form

Given, the equation of a Cartesian circle is given in this general formula: $(x-a)^2 + (y-b)^2 = r^2$. This can be derived from the distance formula and Pythagoras's Theorem. However, how can I derive ...
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0answers
23 views

Prove: The normal to the involute of a circle is tangent to the circle

Please refrain from using algebraic equations (in the Cartesian system) to prove it. I was looking for some kind of geometric proof for it. Or using differentitiation of vectors. For using the ...
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2answers
47 views

Find the area of the hexagon with given conditions [on hold]

Draw a regular hexagon and a regular dodecagon (12-sided polygon) inscribed in a circle. If the area of the dodecagon is $12~\text{cm}^2$, find the area of the hexagon in $\text{cm}^2$. (Express your ...
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1answer
32 views

Proof that the minimum area rectangle is collinear with an edge of the convex hull?

If I have a finite set of points S, is there a way to prove that the minimum area rectangle containing all points in S will be collinear with one of the edges of the convex hull of S? As far as I can ...
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0answers
26 views

Projection from point onto plane

Let the plane is $\prod:\vec{x}\bullet\hat{n}=d$ be a plane, where $\vec{x}\in R^{3}$ and $d\in R$, then can I show using norm and Cauchy-Schwarz inequality that projection of point $x$ onto the ...
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1answer
20 views

Why does a unit vector point in the same direction? [duplicate]

I know how to compute the unit vector $$\hat{\textbf{u}} = \left( \frac{u_1}{||u||} , \dots , \frac{u_n}{||u||} \right)$$ and I also know how to show that this will have length 1 by using the ...
3
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2answers
93 views

Geometry problem related to circle, triangles.

Given acute triangle $\triangle ABC$ satisfying $|\overline{AB}| \ne |\overline{AC}|$. Let $D,E$, respectively, be the midpoints of $\overline{AB}, \overline {AC}$. Let $Q, P$ be the intersections of $...
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1answer
39 views

Existence of connection on dual bundle

I quote the construction given in Madsen's Calculus to Cohomology. This is more or less the construction explained here defining connection on dual bundle. For a vector bundle $\Omega^i(\xi) := \...
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2answers
29 views

How to find orthogonal vector to an arbitrary 3 dimentional vector [duplicate]

Given a vector $\begin{bmatrix}a\\b\\c\end{bmatrix}$ what is a simple solution to find any vector perpendicular to it?
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4answers
48 views

What is the value of $a + b$ if $a\sqrt{b} = BC$ in right triangle $\Delta ABC$?

In $\Delta ABC$, $\angle ABC = 90^\circ$ , $D$ is the midpoint of line $BC$. Point $P$ is on $AD$ line. $PM$ & $PN$ are respectively perpendicular on $AB$ & $AC$. $PM$ = $2PN$, $AB = 5$, $...
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1answer
29 views

Coordinate-free proof that two points are diametrically opposed

Let $c$ be the center of a circle with radius $r > 0$ and let $a$ and $b$ be two points at the circle. If there exists $t \in [0,1]$ such that $c = (1-t)a + tb$, then $d(a,b) = 2r$. I'm working ...
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1answer
22 views

Find value of a with given angle

The equations of the line $L$ and the plane $\Pi$ are as follows: $$ L: \qquad x-5=-(y+1); z = 4 $$ $$ \Pi : \qquad \alpha x + z = 5\alpha +4 $$ where $\alpha$ is a constant. If the angle between ...
2
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0answers
41 views

How to define the complement of a “region” in $\mathbb{R}^d$ using boxes.

I have a question related to set theory, of which I am a beginner. Please add/change tags if you have better references. Other than describing my main question, I'm also highlighting with the symbol [?...
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1answer
44 views

What is a slope of a line?

What is a slope of a line? I understand that it is obtained by tan$\theta$ where $\theta$ is the inclination of the line but overall what does it means?
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1answer
44 views

Prove that in the given pentagon ABCDE AB||CE

In a convex pentagon $ABCDE$ we have $BC\parallel AD$, $CD\parallel BE$, $DE\parallel AC$ and $AE\parallel BD$. Prove that $AB\parallel CE$.
2
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4answers
74 views

Meaning of Dimension

Is a (non-straight) curve in the xy-plane considered to be a 1-dimensional or 2-dimensional entity? Depending on the answer, what is the term used to describe its other attribute (either its oneness, ...
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1answer
41 views

Find number of non-congruent parallelogram of perimeter $40$ with integer side lengths and at least one diagonal of integer length.

Find number of non-congruent parallelogram of perimeter $40$ with integer side lengths and at least one diagonal of integer length. My working: Let sides be $a\le b$ and one diagonal of integer ...
1
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2answers
54 views

Geometric Proof that $\frac{d}{d\theta}(\tan{\theta}) = 1 + \tan^{2}{\theta}$

The following proof sketch is from the preface of Tristan Needham's Visual Complex Analysis: Why is the left side of the black triangle labeled $L \, d\theta$? I can see that the length of this side ...
3
votes
2answers
63 views

Fair Sharing of a Pizza When Opinions About the Edge Differ

Two friends wants to share a pizza. One of them loves the edge of the pizza and the other one hates it. Both consider the pizza to get tastier the closer to the center you get. What is the fairest way ...
3
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0answers
118 views

Poncelet porism for two intersecting circles…

I was already studying about Poncelet porism but unfortunately I couldn't find any useful thing about this theorem for two intersecting circles. even I don't know if it is true for intersecting ...