Questions on the use of algebraic techniques to prove geometric theorems.

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77
votes
20answers
14k views

How to check if a point is inside a rectangle?

There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?
37
votes
9answers
3k views

What is this beauty curve?

Consider the following shape which is produced by dividing the line between $0$ and $1$ on $x$ and $y$ axes into $n=16$ parts. Question 1: What is the curve $f$ when $n\rightarrow \infty$? ...
19
votes
1answer
502 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 ...
18
votes
8answers
5k 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 ...
18
votes
5answers
456 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
1answer
246 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?
15
votes
1answer
622 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
1answer
182 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 ...
11
votes
2answers
409 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
5answers
3k 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
10
votes
2answers
106 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
975 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
3answers
6k 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 ...
9
votes
4answers
172 views

Can we plot a regular octagon on a set of axes, where all vertices of the octagon lie on integer co-ordinates?

I'm a high school teacher and someone asked me this in my class, and to be honest I'm quite stumped! I haven't done any high level math in such a long time, and I'm really not sure how to approach ...
9
votes
2answers
1k 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 ...
9
votes
3answers
274 views

Why do we believe the equation $ax+by+c=0$ represents a line?

I'm going for quite a weird question here. As we know, the equation in Cartesian coordinates for a line in 2-dimensional Euclidean geometry is of the form $ax+by+c=0$. I'm wondering why do we ...
9
votes
2answers
2k 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 ...
9
votes
1answer
190 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
195 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
5answers
198 views

How to think of $\vec{u}-\vec{v}$

Assume I have two vectors, $\vec{u}$ and $\vec{v}$. I know that I can think of their sum via Triangle or Parallelogram Law, but I'm having trouble knowing which way the vector would point depending on ...
8
votes
3answers
995 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
2answers
176 views

The number of grid points near a circle.

There is a circle with center $(0, 0)$ and radius $r$. Let $n$ be the number of grid points inside or on the circle that at least one of its neighboring (up, down, left, right) grid points is outside ...
8
votes
1answer
155 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 ...
8
votes
1answer
366 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
3answers
6k 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
956 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
2k 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
3answers
645 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 ...
7
votes
2answers
511 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 $BP$ ...
7
votes
4answers
142 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 ...
6
votes
6answers
2k 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 $t_2$ ...
6
votes
3answers
618 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
2k 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
1answer
282 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
664 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
765 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 ...
6
votes
0answers
63 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
5
votes
4answers
2k 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
1k 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
5answers
775 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
240 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
2answers
286 views

Generalizing the hardest question on the practice math GRE

The most-missed question on the Math GRE is the following: How many times does $x^{12}$ intersect $e^x$? Because I told you it was hard, you probably realized it was a trick and got the right ...
5
votes
3answers
666 views

What is the difference between vector components and its coordinates?

Some mathematitians told me that vector components and coordinates are different things. They say that vector $F^n$ always has N components but coordinates depend on chosen basis and, therefore, it is ...
5
votes
4answers
1k 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 ...
5
votes
2answers
170 views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
5
votes
1answer
373 views

What are some isometries of $S^2$ without fixed points?

This spherical geometry question involves isometries. I am particularly looking for isometries with no fixed points.
5
votes
1answer
329 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
550 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
91 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
3answers
6k views

Orthogonal projection of a point onto a line

please give me a directions how to solve this: find an orthogonal projection of a point T$(-4,5)$ onto a line $\frac{x}{3}+\frac{y}{-5}=1$