Tagged Questions
0
votes
0answers
40 views
How to prove the property of scalar distribution over vector addition when the vectors are collinear?
$\overrightarrow{a},\overrightarrow{b} \in V^3 , \alpha \in \mathbb{R} $
Prove: $\alpha(\overrightarrow{a} +\overrightarrow{b}) = \alpha\overrightarrow{a} + \alpha\overrightarrow{b}$
When $\alpha = ...
1
vote
2answers
87 views
Finding the locus of the midpoint of chord that subtends a right angle at $(\alpha,\beta)$
There is a circle $x^2+y^2=a^2$. On any line that cuts the circle in two distinct points(it is a secant), the points of intersection with circle are taken and at those two points I draw the tangents ...
1
vote
2answers
98 views
Algebraic solution to find circle radius given distance of three external points from perimeter
I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is ...
0
votes
3answers
52 views
Prove that $|PC|^2 + |PD|^2 = |AB|^2$ if
We have an angle of 90° so that there are 2 points A, B on each side of the angle, O is the vertex and |OA| = |OB|. On the arc AB with it's center being in O, we pick an arbitrary point P and draw a ...
4
votes
1answer
150 views
what are some isometries of S^2 without fixed points?
Spherical geometry question involving isometries.
Particularly looking for isometries with no fixed points.
7
votes
0answers
97 views
Q: Given the graph of $y = \frac{1}{x}$, construct the $(x,y)$ coordinate axes using straightedge and compass
The solution to the problem above is known (see comments for a hint).
What other analytic functions can one substitute for $y = \frac{1}{x}$, and still be able to do so?
0
votes
1answer
30 views
A Statement About Points in the Real Euclidean Space
Suppose that $n \geq 3$, $x$, $y \in \mathbf{R}^n$, $d \colon= |x-y| > 0$, and $r>0$. Then how to prove the following assertions:
(a) If $2r>d$, there are infinitely many $z \in ...
7
votes
4answers
116 views
closest point to on $y=1/x$ to a given point
I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
1
vote
2answers
243 views
Is the area of intersection of convex polygons always convex?
I am interested specifically in the intersection of triangles but I think this is true of all convex polygons am I correct? Also is the largest possible inscribed triangle of a convex polygon always ...
1
vote
1answer
183 views
How to find the intersection of the area of multiple triangles
I have a couple of questions regarding finding the intersection of triangles. I have a system of 16 projectors that all have slightly different color gamuts. The color gamuts are represented by a ...
0
votes
2answers
61 views
Faster Distance formulae for higher n dimension
I need to calculate the distance two points, but these two points contain more than 100 dimensions. With the regular two or three dimension distance formula, we can extend this further to n dimension ...
1
vote
3answers
116 views
Number of ellipses through two fixed points in 2D space?
How many ellipses with a given size (mean $a$ and $b$ given) one can draw through two fixed points in 2D plane?
1
vote
1answer
68 views
Locus perpendicular to a plane in $\mathcal{R}^4$
I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me..
In $\mathcal{R}^4$ I have a plane $\pi$ and a point P. I have to find the ...
2
votes
0answers
224 views
Equations of branches of a mind map
Sorry for the long question, but it's not so simple to explain.
Consider a mind map like this:
I want to draw branches in a cartesian coordinate system.
I'd like to find two equations which ...
9
votes
1answer
172 views
A curve that intersects every plane in finitely but arbitrarily many points
Does there exist a piecewise smooth curve in $\mathbb{R}^3$ such that every plane intersects the curve at finitely many points and the number of intersection points can be arbitrary large?
If the ...
0
votes
4answers
135 views
Figure out if a fourth point resides within an angle created by three other points
If I have a point that is considered the origin and two lines that extend outwards infinitely to two other points, what is the best way to determine whether or not a fourth point resides on or within ...
5
votes
1answer
370 views
Geometry IMO 1988
(IMO 1988/1) Consider two circles of radii $R$ and $r$ $(R > r)$ with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ ...
3
votes
4answers
1k views
Find the area of overlap of two triangles
Suppose we are given two triangles $ABC$ and $DEF$. We can assume nothing about them other than that they are in the same plane. The triangles may or may not overlap. I want to algorithmically ...
1
vote
1answer
2k views
How to find the interior angle of an irregular pentagon or polygon?
I have 5 points and measures of sides of pentagon in 2D. Then how do i find interior angles of pentagon?
Suppose $P_1,P_2,P_3,P_4,P_5$ are five points of Pentagon $P_1P_2P_3P_4P_5$. I know how to ...
1
vote
0answers
220 views
Prove a very original version of Descartes's circle theorem
Prove:
I define the radius of three mutually externally tangent to be $d,e,f$ respectively. The circle with radius $x$ is internally tangent to all three circles. Then
$$ddeeff+ddeexx+ddffxx+eeffxx ...
3
votes
1answer
82 views
Determinant and Measure
The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram.
For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is
\begin{array}{|cc|}
v_1 ...
1
vote
2answers
168 views
Can more than four circles internally tangent or external tangent or combination of both each others at different points?
Is it true for infinite number of m, more than four, there exist m circles internally tangent or external tangent or combination of both each others(in this problem, i mean a circle must be tangent to ...
2
votes
1answer
36 views
Finding unknown 3D vector given 2 known vectors and 2 known angles
So I have a 3D vector math problem that I'm having difficulty solving. Basically I have two known vectors in the form (x,y,z), let's call them C and P, and I want to find a third unknown vector, let's ...
2
votes
1answer
163 views
What is wrong with this proof that isometries must be surjective?
Let $\phi : \mathbb{R}^2\rightarrow\mathbb{R}^2$ be an isometry. Suppose $\phi$ is not surjective, that is there exists some $v \in \mathbb{R}^2$ whose fiber $\phi^{-1}(v)$ is empty. Then by the ...
1
vote
2answers
138 views
Is there a generalized method of rotation for curves?
I know that we can rotate a curve in $R^2$ about a linear axis, as is common for first year calculus problems involving solids of revolution. But has anyone come up with a general method to take a ...
2
votes
4answers
671 views
Proving two lines trisects a line
A question from my vector calculus assignment. Geometry, anything visual, is by far my weakest area. I've been literally staring at this question for hours in frustrations and I give up (and I do mean ...
2
votes
4answers
3k views
Determining if an arbitrary point lies inside a triangle defined by three points?
Is there an algorithm available to determine if a point P lies inside a triangle ABC defined as three points A, B, and C? (The three line segments of the triangle can be determined as well as the ...
5
votes
2answers
162 views
Locus and concurrent lines
This will be my first question :-)
Let $\mathcal{D}_1$ and $\mathcal{D}_2$ two concurrent lines, and $F$ a point in the plane, and $H$ and $G$ its images by the symmetries of axis $\mathcal{D}_1$ and ...
1
vote
2answers
157 views
Distribution or bounds for maximum Cartesian coordinate sampled from the sufarce of an n-sphere
It's been said that for high dimensions a hypersphere is "nearly all equator". The amount of space near the poles is just ridiculously small. This of course means that from a uniformly random sample ...
1
vote
2answers
108 views
Finding a random vector exactly yay far from another point in 3D space
So I am trying to find a vector a certain distance away from another point ( the distance varies based on an input ) and I've figured out that
...
5
votes
6answers
765 views
Product of slopes is -1 iff perpendicular proof from first principles
Once again I'm working through Stillwell's Four Pillars of Geometry. I'm on Chapter 3 where he first introduces coordinates. The question reads,
3.5.1 Show that lines of slopes $t_1$ and $t_2$ ...
3
votes
2answers
93 views
Is this lemma about the minimal distance of two lines true?
In school, I recently proved a solid geometry excercise by assuming that the following lemma is true:
If two lines $g$ and $h$ in the euclidian space are not parallel, and if the lines seem ...
1
vote
1answer
93 views
Converting an angle in Euclid proof-wise into a relation on 'Cartesian' polynomials
In Is it possible to solve any Euclidean geometry problem using a computer? I claimed that one can convert the statement of a theorem in Euclid into multivariate polynomials such that Groebner basis ...
2
votes
1answer
1k views
Getting the third point from two points on one line
My question is the following
how can i get the x3,y3 point from x1,y1 and x2,y2 points?
the x3,y3 point distance from x1,y1 is 300.
1
vote
1answer
372 views
Locus of osculation of concentric ellipses (elliptic pond ripples)
If you dropped two rocks in a pond, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line.
Now imagine that instead of ...
1
vote
1answer
215 views
How close are star-convex sets to convex sets?
What interesting properties of convex sets are retained by star-convex sets?
3
votes
1answer
189 views
Curve of a fixed point of a conic compelled to pass through 2 points
Suppose that in the plane a given conic curve is compelled to pass through two fixed points of that plane.
What are the curves covered by a fixed point of the conic, its center (for an ellipse), its ...
8
votes
3answers
735 views
Minimal Ellipse Circumscribing A Right Triangle
Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one.
You may chose the origin and orientation ...
4
votes
3answers
2k views
Find the coordinates in an isosceles triangle
Given:
A = (0,0)
B = (0,-10)
AB = AC
Using the angle between AB and AC, how are the coordinates at C calculated?
2
votes
1answer
316 views
Collective Term for XY, YZ and ZX planes
Is there a collective term for the XY, YZ and ZX planes in 3D co-ordinate geometry? I was thinking "principal planes" but I'm not sure where I heard that.
