Questions on the use of algebraic techniques to prove geometric theorems.
60
votes
18answers
5k views
How to check if a point is inside a rectangle?
There is a point (x,y), and a rectangle a(x1,y1),b(x2,y2),c(x3,y3),d(x4,y4), how can one check if the point inside the ...
18
votes
5answers
359 views
Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?
When I was in third grade, I was playing with rectangles and diagonal lines, and discovered something very interesting with fractions. I've shown several math teachers and professors over the years, ...
16
votes
6answers
3k views
Is there an equation to describe regular polygons?
For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...
15
votes
1answer
312 views
Intuition why the volume and surface area of the unit sphere eventually decrease
The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$
and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$
both attain maximum values for finite $n$. We can ...
15
votes
1answer
599 views
Bath towel on the rope
This question is related to my bath towel, which I hang on a rope, so let's have fun (you can use your own towel to do this experiment in bath-o).
There is this rectangle with sides $a<b$. The ...
11
votes
2answers
294 views
When do equations represent the same curve?
Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
10
votes
1answer
150 views
algebraic versus analytic line bundles
If one has a quasiprojective complex variety X, there is a natural map from the algebraic Picard group to the analytic Picard group. Is this map either injective or surjective?
I assume the latter ...
10
votes
2answers
83 views
Prove that $|PF_{1}|+|PF_{2}|$ is Constant in an Elipse
Given an elipse with two focus $F_{1}$ an $F_{2}$, and $A$ is an arbitrary point at the elipse. Stright line $AF_{1}$ has another intersection point $B$ with the elipse, and $AF_{2}$ has another ...
9
votes
3answers
632 views
The vertices of an equilateral triangle are shrinking towards each other
For an equilateral triangle ABC of side $a$ vertex A is always moving in the direction of vertex B, which is always moving the direction of vertex C, which is always moving in the direction of vertex ...
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 ...
9
votes
1answer
162 views
Cube skeleton bindings
Imagine that you have a cube skeleton, like so:
Further imagine that you have three rubber bands that you can loop through any of the faces. However, only one rubber band may go through any ...
8
votes
3answers
736 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 ...
8
votes
5answers
1k views
Calculating the area of an irregular polygon
Given the length of the sides of an irregular polygon (no coordinates provided) how do you compute the area of the maximum area of the polygon?
Thanks in advance
8
votes
2answers
1k views
arc-arc intersection, arcs specified by endpoints and height
I need to compute the intersection(s) between two circular arcs. Each arc is specified by its endpoints and their height. The height is the perpendicular distance from the chord connecting the ...
8
votes
1answer
281 views
Navigating though the surface of a hypersphere in a computer game
People in StackOverflow seems not so into this theme, so I thought I could have better luck in here.
I had the idea of an spaceship game where the world is confined in the surface of an 4-D ...
7
votes
2answers
3k views
Finding the intersecting points on two circles
Given 2 circles on a plane, how do you calculate the intersecting points?
In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
7
votes
6answers
652 views
Where can I find Linear Algebra in Nature?
I'm a Computer Science major and I've been studying Analytic Geometry and Linear Algebra this semester. Today my teacher gave a hell of an explanation talking about linear systems, quadratic ...
7
votes
4answers
973 views
What Does Homogenisation Of An Equation Actually Mean?
For example, if we have a conic;
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
What does homogenising this equation with another line (say ax + by + c = 0 ) actually mean? As in, what are the graphical ...
7
votes
1answer
117 views
Maps of $\mathbb{R}^3$ preserving the cross product
Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$:
$$\phi(a \times b)=\phi(a) \times \phi(b)$$
Is $\phi$ necessarily a rotation around the origin or the map ...
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 ...
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?
6
votes
3answers
3k views
Equation of angle bisector, given the equations of two lines in 2D
I have two lines in 2D expressed with general equation (or implicit equation):
First line: $a_1x+b_1y=c_1 \qquad(1)$
Second line: $a_2x+b_2y=c_2 \qquad(2)$
If the two lines are intersecting I will ...
6
votes
3answers
333 views
Parametric form of an ellipse given by $ax^2 + by^2 + cxy = d$
If $c = 0$, the parametric form is obviously $x = \sqrt{\frac{d}{a}} \cos(t), y = \sqrt{\frac{d}{b}} \sin(t)$.
When $c \neq 0$ the sine and cosine should be phase shifted from each other. How do I ...
6
votes
2answers
712 views
Finding shortest distance from point to plane
I need you guys to check my homework question out if I'm wrong or not...
Given point $(1,4,1)$ in need to find the shortest distance between this and the plane $2x_1 - x_2 + x_3 = 5$.
So firstly, I ...
6
votes
3answers
393 views
How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?
I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that
$$
a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3
$$
then the vertices are actually ...
6
votes
1answer
197 views
Why is $m$ used to denote slope?
What is the reason, historically, that the letter $m$ is used to denote the slope of a line?
6
votes
3answers
524 views
What is the area of the portion of 1/8 of an sphere cut off by two parallel planes?
So the problem that I'm trying to solve is as follows:
Assume 1/8 of a sphere with radius $r$ whose center is at the origin (for example the 1/8 which is in $R^{+}$). Now two parallel planes are ...
6
votes
7answers
298 views
Detect when a point belongs to a bounding box with distances
I have a box with known bounding coordinates (latitudes and longitudes): latN, latS, lonW, lonE.
I have a mystery point P with ...
5
votes
4answers
1k views
Why do rhombus diagonals intersect at right angles?
I've looked all over and I can't find a good proof of why the diagonals of a rhombus should intersect at right angles. I can intuitively see its true, just by drawing rhombuses, but I'm trying to ...
5
votes
6answers
746 views
Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?
Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
5
votes
6answers
767 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 $t_2$ ...
5
votes
5answers
514 views
Distance Between A Point And A Line
Any Hint on proving that the distance between the point $(x_{1},y_{1})$ and the line $Ax + By + C = 0$ is ,
$$\text{Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$
What ...
5
votes
3answers
238 views
What are a , b and c?
$$y = ax^2 + bx + c$$
which is tangent at the origin with the line $y=x$, It is also tangential with the line $y=2x + 3$. Determine the function! Draw a figure!
My main question is this solvable? I ...
5
votes
1answer
249 views
What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.
Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
5
votes
2answers
346 views
Find equation of quadratic when given tangents?
I know the equations of 4 lines which are tangents to a quadratic:
$y=2x-10$
$y=x-4$
$y=-x-4$
$y=-2x-10$
If I know that all of these equations are tangents, how do I find the equation of the ...
5
votes
2answers
79 views
If $\left |z-3\right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$
If $\left |z-3 \right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$.
I have got to the point where I have $\left |z \right |= \sqrt{x^2+(y+1)^2} = \sqrt{(x-3)^2+y^2}$
But really ...
5
votes
4answers
374 views
Find extra arbitrary two points for a plane, given the normal and a point that lies on the plane
For a plane, I have the normal $n$, and also a point $P$ that lies on the plane.
Now, how am I going to find extra arbitrary two points ($P_1$ and $P_2$) for the plane so that these three points $P$, ...
5
votes
1answer
498 views
Formula for curve parallel to a parabola
I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
5
votes
1answer
162 views
Trying to find an unknown point just with angles
This is my model:
What I do know:
A, B, C, which form an equilateral triangle
Mab, Mbc, Mac which are the middle points
Angles x and y, which are the angles formed by the segment from the unknown ...
5
votes
2answers
587 views
Tractrix-like curves
Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line?
What is a parametric equation of such ...
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 ...
5
votes
2answers
234 views
The intersections of 2 circles
Lets consider the following (random) question:
Find the intersections of the circles $c_1: x^2+y^2=25$ and $c_2: (x-2)^2 + (y-3)^2=9$
In order to solve this we can do $c_2-c_1$, which leaves us with ...
5
votes
3answers
427 views
A simple(?) Analytical Geometry Question (Ellipse) my teacher can't solve
Here's the story:
I am a high school student who absolutely loves math. So I took a university level mathematics course that is renowned throughout our school for being extremely rigorous and tough. ...
5
votes
3answers
320 views
Why do all circles passing through $a$ and $1/\bar{a}$ meet $|z|=1$ are right angles?
In the complex plane, I write the equation for a circle centered at $z$ by $|z-x|=r$, so $(z-x)(\bar{z}-\bar{x})=r^2$. I suppose that both $a$ and $1/\bar{a}$ lie on this circle, so I get the equation
...
5
votes
0answers
94 views
Proper mapping theorem
My professor mentioned a proper mapping theorem after the name of Remmert which says:
Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex ...
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 $BP$ ...
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?
4
votes
4answers
762 views
How to find the distance between a point and line joining two points on a sphere?
How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below.
In the above ...
4
votes
2answers
631 views
Argument of the sum of two complex numbers
Let $r$, $s$ be positive real numbers and $\theta$, $\phi$ real numbers with $|\theta -\phi|<\pi$. Then an argument of $re^{i\theta}+se^{i\phi}$ lies between $\theta$ and $\phi$.
Can someone give ...
4
votes
1answer
1k views
Proving the Shoelace Method at the Precalculus Level
Using only precalculus mathematics (including that the area of the triangle with vertices at the origin, $(x_1,y_1)$, and $(x_2,y_2)$ is half of the absolute value of the determinant of the $2\times ...
