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

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3
votes
2answers
280 views

line equidistant from two sets in the plane

Suppose $A$ and $B$ are disjoint subsets of the plane, both closed, nonempty, and connected. Define $E(A, B)$ as the set of points in the plane equidistant from $A$ and $B$. For example, if $A$ is a ...
11
votes
3answers
2k views

Construction of a right triangle

It's a high school level question which we can't seem to solve. Here it is: Given 2 lines, one of the length of the hypotenuse and the other with the length of the sum of the 2 legs, construct ...
4
votes
3answers
370 views

given an altitude and a viewing angle how do I determine the distance of the viewing area

What is the formula to solve this problem. Say I have a tower 100km tall and I want to determine the distance from the base of the tower to where a cable is attached to the ground. The cable forms a ...
0
votes
1answer
59 views

Matrix for transform line into y-axis

I have a line with equation $x_i = a_i t + b_i$, for $i = 1, 2, 3$ (if such way not good i can use any other) with which matrix i can transform this line into $y$-axis? I need to do polenty of ...
1
vote
1answer
308 views

map on a unit sphere with polar coordinates

My brother, who is in hospital atm and cannot verify by himself asked me to post the following question, thank you in advance, and sorry if the topic has already been covered, i do not have the math ...
1
vote
2answers
71 views

Algebra of Vectors

Is it possible that there are $3$ vectors, $a, b, c$, such that $a + b + c = 0$ but $|a| = 1$, $|b| = 2$ and $|c| = 4$? If yes why? and if no why?. I'm trying to get the solution since last $2$ ...
9
votes
3answers
336 views

Triangle problem

I have got one simple task to prove: We have got a triangle $\triangle XYZ$. Then we create points $A,B,C$ on $XY, YZ, ZX$ respectively, such that $XA = AB = BZ$ and $CZ = AY = AC$. How to prove ...
3
votes
1answer
2k views

Average Distance Between Random Points in a Rectangle

My question is similar to this one but for rectangles instead of lines. Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed ...
37
votes
11answers
4k views

What is a hexagon?

Having a slight parenting anxiety attack and I hate teaching my son something incorrect. Wiktionary tells me that a Hexagon is a polygon with $6$ sides and $6$ angles. Why the $6$ angle requirement? ...
2
votes
2answers
631 views

Pattern matching circle, square or triangle

I have a set of x, y co-ordinates that are actually taken from hand drawings of a circle, square or a triangle. Using the set of points, I need to mathematically find if the points approximately fit a ...
1
vote
1answer
127 views

Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb H3$, has a boundary $\mathbb CP1$. A ideal tetrahedron in $\mathbb H3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb CP1$. The four vertices ...
1
vote
2answers
1k views

Is this a wrong solution to the smallest enclosing circle problem?

I have a set of points in $\mathbb{R}^2$ and I need to find the smallest enclosing circle, i.e. the circle with the smallest radius that contains all of the points belonging to the set. I have the ...
2
votes
1answer
97 views

Vector Algebra, Addition of vectors

Is it possible that 3 vectors, $\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} =0$ but $|\overrightarrow{a}|$, $|\overrightarrow{b}|$ and $|\overrightarrow{c}|$ do not represent the ...
1
vote
1answer
191 views

How do I calculate the volume of the intersection of a cylinder with a not concentric sphere?

I have, in metal, cut a $1.5''$ radius hemisphere (left half); now I want to cut a $1.8''$ diameter cylinder at $120^\circ$ from the horizontal and $0.333''$ deep on one edge and $0.031''$ deep at the ...
2
votes
1answer
74 views

Number of triangles

In a triangle $\bigtriangleup ABC$ is $\widehat A=30^{\circ}$, $|AB|=10$ and $|BC|\in\{3,5,7,9,11\}$. How many non-congruent trangles $\bigtriangleup ABC$ exist? The possible answers are $3,4,5,6$ ...
2
votes
2answers
81 views

One question on distances in an equilateral triangle

I am self-studying Euclidean Geometry, and I want to solve the following exercise. Let $ABC$ be an equilateral triangle with height $h$, and $P$ is a point in its interior. If $x,y,z$ denote the ...
0
votes
1answer
2k views

Parametric and implicit representation of a cone

http://mathworld.wolfram.com/Cone.html shows the parametric and implicit representation of a cone, I am wondering what the equation would look like if we also consider the bottom circle face for the ...
2
votes
2answers
727 views

How to find the minimal axis-parallel ellipse enclosing a set of points.

I have a set $X$ of points in $\mathbb{R}^2$ and I'm trying to find the smallest encompassing ellipse which main axes are parallel to the coordinate system's (to put it differently, its both centres ...
2
votes
1answer
533 views

Embedded Submanifold

This is a question from Lee : Introduction to Smooth manifolds. p.201 For each $a \in \mathbb{R}$, let $M_a$ be the subset of $\mathbb{R}^2$ defined by $$M_a = \{(x,y) : y^2 = x(x-1)(x-a)\}$$ For ...
1
vote
2answers
1k views

Why can two non-overlapping circles intersect in at most two points, while two non-overlapping ellipses can intersect at four?

When reading about why no Venn diagram for four sets can be formed by intersecting four circles, I found that the author claimed that any two distinct circles can intersect in at most two points, ...
4
votes
1answer
91 views

Challenge question for normed linear spaces.

I have come across the following challenging problem in my analysis course: Let $K$ be a compact convex set in a normed linear space. Suppose that $$\sup_{x,y\in K}\{||x-y||\}=\delta>0.$$ Show ...
1
vote
1answer
110 views

$\sqrt{A(ABCD)} =\sqrt{A(ABE)}+ \sqrt{A(CDE)}$

I am self-studying Euclidean Geometry, and I want to solve the following exercise. Let $ABCD$ a trapezoid with bases $AB, CD$, and $AC, BD$ are the lateral sides. If $E$ is the point of intersection ...
2
votes
1answer
645 views

Intersection of Two Circles

I have two circles as: $C_1: (x-x_1)^2+(y-y_1)^2=r_1^2$ and $C_2: (x-x_2)^2+(y-y_2)^2 =r_2^2$ and these circles have non-empty intersection. In other words $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\leq ...
3
votes
2answers
571 views

Similarity involving Miquel's Theorem

Let $\Delta ABC$ be a triangle. If we place points $D,\ E,\ F$ arbitrarily on the sides $\overline{AB},\ \overline{BC}$ and $\overline{CA}$ respectively, then the circumcircles of the triangles ...
0
votes
1answer
221 views

How to prove euler formula for surface meshes with disk or sphere topology?

For disk topology the euler formula is V - E + F = 1, for sphere it is V - E + F = 2. Is there a simple and elegant way to prove these?
1
vote
1answer
738 views

Relation: average (squared) distance to all points, and (squared) distance to centroid

Suppose a set of $n$ high-dimensional points is given. It is known that the sum of all pair-wise squared Euclidean distances is proportional to sum of squared distances of all points to the centroid. ...
4
votes
1answer
350 views

Convex combination

Assume that $I$ is a countable set, and we have $u_i\in \mathbb{R}^n$ for $i\in I$. Suppose that $v=\sum_{i\in I} a_i u_i$ and $\sum_{i\in I}a_i=1$ and $a_i\geq 0$. Can one show that there exists a ...
0
votes
2answers
688 views

Rectangular problem

I was trying to solve this problem: Let P be a point in the interior of rectangle ABCD. Given PA = 3, PD = 4 and PC = 5, find PB. I feel lost because it's not right to assume P is in the center ...
3
votes
4answers
977 views

Is “$n$ is divisible by $4$ if and only if $n^2$ is even” a True Statement?

I'm working on some high school geometry homework, and I'm having some trouble with a problem about proofs and counterexamples. The question posses the statement $n$ is divisible by $4$ if and only ...
1
vote
1answer
65 views

Determine the path that created by a sector of circle

ABC circle sector turns on ground (x axis) as shown in the figure. A is the center of the circle. $\angle{OAB}=\angle{OAC}=\alpha $ $|AB|=r$ $\cfrac{|AP|}{|PC|}=k$ The corners meet on point $H$ ...
1
vote
3answers
2k views

Dividing circle into six equal parts and know the coordinates of each diving point …

I have a circle who center $(0, 0)$ and radius $100$ are known. That circle is divided into $6$ equal parts. I want to know the coordinates of all six points on the circle that divides it into $6$ ...
1
vote
1answer
102 views

Is the following algorithm to check if a point is on a line correct?

I need to check if a point is inside a rectangle (which can be not parallel to the axes) but not on its perimeter. In 2 dimensions. I'm checking if the point is inside by finding the intersection ...
0
votes
1answer
325 views

Asymptotes of a hyperbola

Is this the correct solution something doesnot feel right
1
vote
2answers
243 views

coordinates of the point where 2 tangents to a circle cross

I have a circle of radius r. Given two lines tangent to the circle at points (x1,y1) and (x2,y2), What are the coordinates of the point where the two tangents cross?
0
votes
1answer
59 views

Eccentricity of a conic

I got this solution, is this right?
1
vote
1answer
67 views

Calculate the angle of a rotated conic?

I am required to calculate the rotation angle needed to come into standard form without x y product term (to make axes parallel to conic axes) in trying to find solution of problem: A conic $M$, ...
3
votes
1answer
603 views

Find the eclipse focal point

A conic with equation $$ a x^2 + b y^2 = c $$ has two focus points, where $a=4$, $b=24$ and $c=65$. One of those focus points has a positive x-coordinate. To two decimal places, what is the ...
0
votes
0answers
40 views

A good way to embed a manifold in a Euclidian space $\mathbb{R}^n$

We know that any closed manifold $X$ can be embedded into some Euclidian space $\mathbb{R}^n$ for sufficiently large $n\in \mathbb{N}$. What is the easiest way to see this fact? I have seen several ...
1
vote
0answers
163 views

are oblique projections one specific subdivision of trimetric projections?

So I've reaserched a while and come with this broad definitions a projection is the representation of a 3D object in 2D by the use of "imaginary proyectors"(cameras of some sort). it has 2 branches, ...
0
votes
1answer
33 views

Need help developing the formula to calculate the length of the y axis of a right triangle with a curved side for any position on the x axis.

If a right triangle has one side that is 500, another side that is 208, and the last side with a radius of 705, what is the formula to determine the length of the intersection point (y) at any given ...
1
vote
1answer
603 views

Implicit surface representation of a cube

Implicit representation is of the form f(x, y, z) = 0. For a sphere it is just $x^2 + y^2 + z^2 = R^2$. I am wondering what is the equation for a cube.
1
vote
1answer
310 views

An ellipse in the rhombus

Suppose that there is an ellipse that meets with the square, but exactly inside the rhombus. The rhombus's side would be some $x$ cm. (for e.g., we can take it as $2 \ cm$.) The ellipse would have a ...
1
vote
1answer
120 views

Finding the x- coordinate in triangle

Is it possible to find the point which is marked by question mark ? we know that the s1(x)=s2(x) (the areas of the two triangles are equal)
2
votes
1answer
279 views

Graph Isomorphisms, Delaunay Triangulation on a sphere, and Kulikowski's Theorem

Suppose I have a collection of $n$ non-collinear points on a sphere, $\left\lbrace P_i\right\rbrace_{i=1}^n $. And I construct a mapping from this collection of points to the Delaunay Triangulation ...
1
vote
0answers
44 views

Uniformly sample points over a circular patch of a sphere without rejection [duplicate]

Possible Duplicate: Generate a random direction within a cone A point on a unit sphere $(x,y,z)$ and an maximal angular separation $\theta$ defines a patch with an area of $\Omega = 2 \pi ...
1
vote
4answers
391 views

Area of Infinite Shaded Regions

I recently came across this WISCONSIN STATE MATHEMATICS MEET problem and solution. The solution reads that one of the larger triangles has an area of 1/16 th of whole square. How is that?
3
votes
3answers
446 views

Level curves on ellipsoid

Let $a,b,c>0$ with $a\leq b\leq c$. Let $E$ be the ellipsoid determined by $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$$ Is there a function $f:E\rightarrow \mathbb{R}$ such ...
2
votes
1answer
90 views

How to find a point P in △ ABC,△ PAB,△ PBC, the△ PCA inscribed circle radius are equal?

How to find a point P in △ ABC, △ PAB,△ PBC, the△ PCA inscribed circle radius are equal?
-1
votes
3answers
2k views

area of a convex quadrilateral

I have a quadrilateral with sides as follows: $30, 20, 30, 15.$ I do not have any other information about the quadrilateral apart from this. Is it possible to calculate its area?
3
votes
1answer
209 views

Computing the trajectory of an orbiting body so that it collides with another orbiting body

I am creating a 2D game in which two space ships, orbiting around a planet under the influence of gravity, fire projectiles at each other, which are also under the influence of gravity. I'm creating ...