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

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4
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1answer
340 views

10 points inside a square - minimum distance between any of them

A square of side 1 is given, and 10 points are inside the square. If we divide the square into 9 smaller squares, and apply Dirichlet principle, we can prove that there are 2 of these 10 points whose ...
0
votes
1answer
519 views

How to sum 2 vectors in spherical coordinate system

I have 2 vectors, and each defined by 3 coordinates: radius, azimuth and zenith How i can sum it? Help me pls guys. What i have done: I found azimuth and zenith of resultant vector, by decomposition ...
1
vote
1answer
204 views

Reference - Fractal Geometry

I am looking for textbooks or lecture notes about Fractal Geometry that reach an advance level on the topic and aren't just introductory.
1
vote
1answer
103 views

Please, as possible, explain in layman's terms: What is a discontinuous space?

What is a "discontinuous space"? Is it synonymous of "discrete space"? I searched in Google but did not find an accessible explanation. I have an idea of it as a space where all lengths are multiples ...
0
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2answers
339 views

Image of a point reflected over $y=mx+b$ using dot product

So, I know that the image for a generic point is $$\left(\frac{1-m^2}{1+m^2}x + \frac{2m}{1+m^2}(y-b), \frac{2m}{1+m^2}x - \frac{1-m^2}{1+m^2}(y-b)+b\right)$$ when you reflect it over the line ...
0
votes
1answer
511 views

deriving formula for reflection over y=mx+b using dot product

So, I know that the formula for a generic point is $$\left(\frac{1-m^2}{1+m^2}x + \frac{2m}{1+m^2}(y-b), \left(\frac{2m}{1+m^2}\right)x - \left(\frac{1-m^2}{1+m^2}\right)(y-b)+b\right)$$ when you ...
0
votes
2answers
47 views

Affine transformation $f$ such that $f(S^1)\subset S^1$

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be an affine (non trivial) transformation ($f(X)=AX+b$ with $A\neq 0$) such that $f(S^1)\subset S^1$ (i.e. the unit circle is $f$-invariant). Prove ...
2
votes
2answers
229 views

Find the side length of a hexagon

In a convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles. And $\angle B$, $\angle C$, $\angle E$ and $\angle F$ are congruent. The area of the hexagonal ...
0
votes
0answers
25 views

Which bound on $a,b,c,d$ is correct?

Let $ABCD$ be a unit square. Four points E,F,G, and H are chosen on the sides $AB,BC,CD,$ and $DA$ respectively. Let the length of the quadrilateral be $a,b,c,d$. Then which of these is always true? ...
2
votes
2answers
53 views

Finding the equation of tangent line

I'm stuck with problem supposed to be trivial. I need to find tangent line witch touches curve $y^2 = -4ax$ at the point $(x_0,y_0)$ Rewriting it as $$x = -\frac {y^2}{4a}$$ Taking derivative: ...
2
votes
1answer
56 views

What does $E^d$ mean?

I was reading the paper "Cutting Hyperplanes for Divide-and-Conquer" by B. Chazelle and in the introduction I came across the following: "Let $H$ be a set of $n$ hyperplanes in $E^d$." What does $E^d$ ...
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2answers
20 views

HELP: find the type of a conic from the given equation

However, I am not sure what conic type it is. Should it be divided by 4 in order to get a standard form of a hypebola? Any help will be appreciated. Thank you!
0
votes
1answer
82 views

Area of a square inscribed in a triangle?

Construct a triangle ABC whose base AB is 24 and altitude CH is 16. Hence, inscribe a square EPGF (one of whose sides lies along AB) and calculate its area. My book suggests to use triangle ...
2
votes
1answer
118 views

Horses grazing in a circle.

Question: Diagram: Note that The circle with center $C$ is touching the arc of semi-circle $AB$ also; I couldn't draw it. The figure wasn't drawn on cartesian planes; so, though it may seem ...
5
votes
1answer
298 views

Geometrical question just for fun

Was puzzling with the following (home made) puzzle: Given the square $ABCD$ with $A = (1,1)$, $B = (1,-1)$, $C = (-1,-1)$ and $D = (-1,1)$ And given point $E = (0,2)$ What is the smallest (by ...
0
votes
2answers
122 views

Is a ray actually a half-line?

I just started studying elementary geometry with Kiselev's plane geometry book. In §5 of the introduction, the author talks about rays, calling them 'half-lines'. That got me wondering whether an ...
3
votes
1answer
97 views

Point from three circle chords

For $i=1,2,3$, let $C_i$ be the circle in the $xy$ plane with center $\mathbf{c}_i = (x_i,y_i)$ and radius $r_i$. Assume that these three circles all intersect one another (6 intersection points, in ...
0
votes
0answers
121 views

Ratio of area of triangle to that formed by its medians

What is the ratio of the area of a triangle $ABC$ to the area of the triangle whose sides are equal in length to the medians of triangle $ABC$? I see an obvious method of brute-force wherein I can ...
1
vote
4answers
111 views

Areas in a rectangle

Suppose $P,Q, R$, and $S$ are the midpoints of the sides $AB, BC, CD$, and $DA$, respectively of rectangle $ABCD$. If the area of the rectangle is $\delta$, then prove that the area of the figure ...
1
vote
1answer
32 views

Furthermore fair dice?

In the field of board games, it is immediately apparent that fair die can be constructed for 2, 4, 6, 8, 10, 12 and 20 sides; represented by the coin, tetrahedron, cube, octahedron, decahedron, ...
2
votes
1answer
77 views

Geometry, need to find angles, area and side lengths on a paralellogram

I have no idea how to solve this, as it is kinda beyond what I have already learned. It goes like this: $M$ is the diagonal's point of intersection (or center I guess) in the parallelogram $ABCD$. ...
0
votes
0answers
77 views

Variations of transformation of inversion?

Is there a transformation analogous to inversion, that is based on something other than circle (or sphere in higher dimensions), and has some interesting properties or applications? The motivation ...
0
votes
1answer
16 views

Correctness of the size of an planar integer lattice unknot

A planar integer lattice unknot is a polygon drawn over a two dimensional integer lattice. Here is an example: Given a number $N$, a planar unknot is not always possible. For example, a planar ...
0
votes
1answer
33 views

How to find the intersection of three given planes

Let $$ a_1 x+b_1 y+c_1 z=d_1\\ a_2 x+b_2 y+c_2 z=d_2\\ a_3 x+b_3 y+c_3 z=d_3 $$ be three planes and it was given that if $d_1=d_2=d_3=1$ then the planes intersect exactly one point. Now my ...
0
votes
1answer
38 views

How to prove that given a line L prove that all points of a fixed distance k form two lines parallel to L

How can I prove start this?I know intuitively since they never meet they are parallel, but I don't think that is a direct proof.
0
votes
1answer
262 views

Barycenter of a tetrahedron

Given a tetrahedron whose vertices are represented as vectors $v_A,v_B,v_C,v_D$, I have the following questions: 1) How to represent its barycenter $v_E$? 2) Are ...
1
vote
1answer
69 views

Orthocentre of triangle and related ratio

$ABC$ is a triangle with $AB = 13$, $BC = 14$ and $CA = 15$. $AD$ and $BE$ are the altitudes from $A$ to $B$ to $BC$ and $AC$ respectively. $H$ is the point of intersection of $AD$ and $BE$. Then the ...
3
votes
1answer
57 views

find the point $P$ such that the expression has minimum value

Let $ABC$ be a triangle with sides $$a,b,c.$$ Find a point $P$ inside the triangle such that $$a(PA)^2+b(PB)^2+c(PC)^2$$ is minimum
0
votes
1answer
44 views

Show that the area of the triangle ABC is maximized when $\angle BCA$ = $\angle CAB$

Let A, B, and C be three points on a circle of radius 1. Suppose that the magnitude of $\angle ABC$ is fixed. Then show that the area of the triangle ABC is maximized when $\angle BCA$ = $\angle ...
2
votes
1answer
393 views

Finding the missing coordinate of a point within a 3D triangle

We have an equilateral triangle $ABC$ in 3-dimensional space. The points are known, such as: $A = (x_1,y_1,z_1)$ $B = (x_2,y_2,z_2)$ $C = (x_3,y_3,z_3)$ Point $P$ is on triangle $ABC$. If I know ...
2
votes
1answer
43 views

Determine the length of **DC** in terms of $l_1$ and $l_2$

In the given figure, E is the midpoint of the arc ABEC and ED is perpendicular to the chord BC at D. If the length of the chord AB is $l_1$, and that of BD is $l_2$, determine the length of DC in ...
4
votes
1answer
264 views

How does Schwartz's paradox of surface area effect modelling of 3D objects?

Question I just became aware of Schwartz's paradox of surface area (explanation below for the unfamiliar). How does this effect mathematical modelling of real-life surfaces? For example, suppose I ...
0
votes
1answer
62 views

The largest possible number of intersections between $n$ lines is $n(n-1)/2$

Let $C_n$ be the largest possible number of intersection points of a family of $n$ lines in the plane. Prove that $C_n = n(n-1)/2$ (If some lines are parallel, or if three lines intersect at a single ...
2
votes
0answers
35 views

integrate the square of angular distance from the node of a spherical triangle

Guessab, Noouisser, and Schmeisser "A Definiteness Theory for Cubature Formulae of Order Two", Constructive Approximation (2006)24:263-288 Define a quantity $R[||\cdot||^2]$ which is $$\sum_{i=1}^N ...
0
votes
1answer
42 views

Pushing a line segment in parallel - calculating new end points

Say you have a line segment in 2d plane. What I want to do is push this line segment in parallel by length of the segment and find out where A and B end up. Here's a picture to make it clearer. ...
0
votes
1answer
87 views

Regular Parametrization of a Sphere

Is there a function $f:U→ \mathbb{R^3}$, such that: (1) U is an open connected subset of $ \mathbb{R^2} $; (2) f is $ C^r , r≥1$; (3) the Jacobian of f is of maximal rank at all points of U; (4) ...
3
votes
2answers
163 views

Finding the angle between the $2$ radii of a circle

Consider a circle with centre $O$. Two chords $AB$ and $CD$ extended intersect at a point $P$ outside the circle. If $\angle AOC = 43^\circ$ and $\angle BPD = 18^\circ$, then what is the value of ...
4
votes
1answer
51 views

$\sqrt{\frac{15}4+\sum\cos(A-B)}\ge\sum\sin A$ in a triangle?

How can I prove that ( $\small{\sum}$ denotes cyclic sum here), for any triangle $ABC$: $$\sqrt{\frac{15}4+\sum\cos(A-B)}\ge\sum\sin A$$ I don't see where to begin even. Any hints would be ...
1
vote
1answer
67 views

Knowing the longitude and latitude of a lunar crater, can I calculate how close to the centre of the disk it will appear to be?

I am seeking a formula to enable me to take the latitude and longitude of a lunar crater (lots of them actually) and turn that into a percentage figure of the distance from the centre of the disc to ...
3
votes
1answer
149 views

Why does $\arctan(\frac{\tan \theta}{2}) \approx \frac{1}{2 - \theta} - \frac{1}{2 + \theta}$ for small $\theta$?

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the ...
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vote
2answers
364 views

Calculating Intersection of Three Spheres Step by Step

How do I calculate the intersection of three spheres step by step? Assume that the spheres are $S_i(c_i, r_i)$ where $i = 1,2,3$, $c_i$ is the center coordinates of $S_i$ and $r_i$ is the radius of ...
1
vote
1answer
65 views

How many different right triangles are possible with the shorter side of odd length?

I was trying to solve this problem but unable to figure it out completely. I thing number of was odd integer $n$ can be the side of right triangle is number of factor of $\frac{n^2}{2}$. Can some one ...
2
votes
2answers
359 views

Is there a function that draws a triangle with rounded edges?

Gabriel Lamé formula allows to convert a circle to a rounded rectangle and finally to a rectangle: $$ x^n + y^n=1 $$ Is there a formula from which to connect the triangle to a rounded triangle to a ...
1
vote
1answer
747 views

Application of mid-point theorem

Consider convex quadrilateral $ABCD$. Let there be a point $P$ in the interior of the quadrilateral such that $PA = PB$ and $PC = PD$. $K,L, M$ are the mid-points of $AB , BC , CD$ respectively. ...
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vote
1answer
182 views

Geometric Intuition Of Partial Derivative in 2D space

$$f(x,y)=3x^2+5y^2-2x+3y+7=0\\f'x=6x-2=0\\x=\frac {1}{3}\\f'y=10y+3=0\\ y=\frac {-3}{10} \\ O(\frac {1}{3},\frac {-3}{10})$$ As you see, we've used the partial derivative to find the center point of ...
2
votes
2answers
74 views

Interior point of $\Delta\,ABC$

if $P(\lambda,2)$ is an interior point of $\Delta\,ABC$ formed by the lines $$x+y=4$$ $$3x-7y=8$$ $$4x-y=31$$ Find $\lambda$ My Idea: The vertices of $\Delta ABC$ are $A(\frac{18}{5},\frac{2}{5})$ ...
1
vote
1answer
1k views

Finding the locus of a mid-point

Let $A$ be the fixed point $(0, 4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $R$. Find the locus of the ...
3
votes
2answers
4k views

Given perimeter of triangle and one side, find other two sides

In triangle ABC, all three sides have integer lengths. If AB = 21, the perimeter is 54, and the area is a positive integer, what are the lengths of BC and AC? I tried using Heron's Formula, but I ...
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vote
2answers
56 views

4-dimensional object with 5 vertices enclosing the origin

A triangle is a 2-dimensional shape with 3 vertices that may be positioned to enclose the origin. A triangular pyramid is a 3-dimensional shape with 4 vertices that may be positioned to enclose the ...
0
votes
1answer
68 views

The Area Of The Circle

If we consider a circle with center O, and a point on the circle say P.The radius will be the vector OP.Now, if we are to consider its area why don`t we multiply the radius OP by the perimeter of the ...