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

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30
votes
9answers
13k 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 ...
111
votes
22answers
47k 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?
9
votes
6answers
21k views

Get the equation of a circle when given 3 points

Get the equation of a circle through the points $(1,1), (2,4), (5,3) $. I can solve this by simply drawing it, but is there a way of solving it (easily) without having to draw?
11
votes
13answers
18k views

Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the ...
15
votes
4answers
2k views

What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
8
votes
3answers
3k views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
1
vote
2answers
12k views

How to find shortest distance between two skew lines in 3D?

If given 2 lines $\alpha$ and $\beta$, that are created by 2 points: A and B 2 plane intersection I want to find shortest distance between them. $$\left\{\begin{array}{c} P_1=x_1X+y_1Y+z_1Z+C=0 ...
10
votes
7answers
535 views

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
7
votes
1answer
4k views

Analogue of spherical coordinates in $n$-dimensions

What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by ...
19
votes
3answers
932 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 ...
0
votes
4answers
268 views

Equal perimeter and area

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
9
votes
2answers
11k 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 ...
6
votes
3answers
3k 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 ...
2
votes
5answers
12k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
1
vote
1answer
323 views

Finding equation of an ellipsoid

Consider I have an ellipsoid (let say an egg) lies in a general form in 3D space. Suppose, I have the equations of two projected views of this egg (e.g. one projected view on x-y plane and another one ...
3
votes
4answers
9k views

Common tangent to two circles

Find the equations of the common tangents to the 2 circles: $$(x - 2)^2 + y^2 = 9$$ and $$(x - 5)^2 + (y - 4)^2 = 4.$$ I've tried to set the equation to be $y = ax+b$, substitute this ...
10
votes
3answers
6k views

The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
10
votes
3answers
13k 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 ...
4
votes
2answers
3k views

A good Open Source book on Analytic Geometry?

Hi my course specifically talks about : Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, ...
3
votes
2answers
6k views

How do we prove the rotation matrix in two dimensions not by casework?

I was trying to prove: To carry out a rotation using matrices the point $(x, y)$ to be rotated from the angle, $θ$, where $(x′, y′)$ are the co-ordinates of the point after rotation, and the formulae ...
9
votes
1answer
547 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 ...
3
votes
5answers
652 views

Help understanding cross-product

I am trying to calculate the intersection point (if any) of two line segments for a 2D computer game. I am trying to use this method, but I want to make sure I understand what is going on as I do it. ...
12
votes
5answers
23k views

What is the general equation of the ellipse that is not in the origin and rotated by an angle?

I have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?
5
votes
1answer
210 views

Tensor notation (practicing)

I'm praticing tensor notation, and I want to prove this way that given vectors $A,B,C,D$ then $(A \times B) \times (C \times D) = \det(A,C,D)B - \det(B,C,D)A$, where $\det$ means the triple product. ...
15
votes
4answers
758 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 ...
10
votes
3answers
2k 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 ...
8
votes
2answers
2k views

implicit equation for “double torus” (genus 2 orientable surface)

The embedded torus in $\mathbb R^3$ can be described by the set of points in $(x,y,z)\in \mathbb R^3$ satisfying $T(x,y,z)=0$, where $T$ is the polynomial ...
4
votes
2answers
159 views

How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
3
votes
1answer
665 views

Enlarging an ellipses along normal direction

Given an ellipses, enlarge it along normal direction a fixed length say 1cm. Do we get another ellipses? If so, how to prove ?
11
votes
4answers
4k 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 ...
5
votes
1answer
95 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
3
votes
1answer
212 views

Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$

Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$ The required condition is $a^2>8b^2$.I dont know how ...
3
votes
1answer
311 views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
1
vote
3answers
66 views

When do the linear combinations of three vectors fill out the triangle?

Let $P$, $Q$, and $R$ be any three (non-collinear) points in the 3D space. Let $O$ denote the origin. Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ denote the vectors $\vec{OP}$, $\vec{OQ}$, and ...
0
votes
2answers
51 views

What is the coordinate of a point $P$ on the line $2x-y+5=0$ such that $|PA-PB|$ is maximum where $A=(4,-2)$ and $B=(2,-4)$

What is the coordinate of a point $P$ on the line $2x-y+5=0$ such that $|PA-PB|$ is maximum where $A=(4,-2)$ and $B=(2,-4)$. Let the coordinates of the point $P$ be $(x_1,y_1).$ $P$ lies on the ...
0
votes
3answers
129 views

Convert this equation into the standard form of an ellipse

$$\frac{\left(\frac{xa^2}{a^2y^2+\ x^2}-p\right)^2}{a^2}+\left(\frac{ya^2}{a^2y^2+\ x^2}-q\right)^2=k^2$$ Could someone please convert this into standard form of equation ...
11
votes
5answers
4k 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
3answers
1k 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 ...
5
votes
4answers
2k 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
356 views

Focal length of an ellipse and related results

There are 2 questions(part of same question but I divided it into two): Q1. Prove that the length of the focal chord of the ellipse $\frac {x^2}{a^2}+\frac {y^2}{b^2}=1$ which is inclined to the ...
2
votes
4answers
2k 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 ...
1
vote
1answer
287 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
0
votes
2answers
37 views

Not coplanar points in $\mathbb{R}^3$ if and only if its determinant is nonzero [closed]

How to prove that points $P_1(x_1,y_1,z_1)$, $P_2(x_2,y_2,z_2)$, $P_3(x_3,y_3,z_3)$, $P_4(x_4,y_4,z_4)$ are not coplanar if and only if the determinant $$\left|\begin{array}{cccc} ...
0
votes
1answer
72 views

Can any smooth planar curve which is closed, be a base for a 3 dimensional cone?

A cone in 3 dimensions has a vertex and a base. The contour of the base is a circle which is a smooth closed planar curve. Can there be a more general cone which can have any smooth closed planar ...
5
votes
3answers
1k 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. ...
4
votes
1answer
749 views

How to calculate the area closed by a parabola and a line without calculus?

In order to simplify the problem, suppose we have a parabola $y=ax^2+bx+c$, here $a\neq0$, and a line $y=kx+d$, here $k\neq0$. We can assume that they will intersect at two different points. Thus, the ...
3
votes
2answers
410 views

Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings?

A book on CG says: ... we can construct any affine transformation from a sequence of rotations, translations, and scalings. But I don't know how to prove it. Even in a particular case, I found ...
2
votes
1answer
64 views

What geometric object is given by this equation?

What geometric object is given by this equation? $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z-6=0$ Maple says it's a hyperboloid of one sheet, but is there a way to show it without going the long way by using the ...
2
votes
4answers
9k views

How to Determine an Equation of a Circle using a Line and Two Points on a Circle

My question goes like this: Determine the equation of a circle tangent to the $x$-axis and passing through $(5,1)$ and $(12,8)$. I need not only the answers, but also the steps on how you did it so ...
2
votes
3answers
289 views

How to obtain equation of line of the form $ax + by + c = 0$?

I'm trying to check if a line hits a rectangle, and for that, I found this nice solution: Line triangle intersection The problem is that, having forgot almost all I ever knew about math, I don't ...