Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.

Analytic geometry, also called coordinate geometry, mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry.

The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra, and vice versa; the methods of either subject can then be used to solve problems in the other.

Analytic geometry was introduced by René Descartes in $1637$ and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late $17^{th}$ cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry.

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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?
Freewind
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Prove the theorem on analytic geometry in the picture.

I discovered this elegant theorem in my facebook feed. Does anyone have any idea how to prove? Formulations of this theorem can be found in the answers and the comments. You are welcome to join in the discussion. Edit: Current progress: The theorem…
user122049
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How to straighten a parabola?

Consider the function $f(x)=a_0x^2$ for some $a_0\in \mathbb{R}^+$. Take $x_0\in\mathbb{R}^+$ so that the arc length $L$ between $(0,0)$ and $(x_0,f(x_0))$ is fixed. Given a different arbitrary $a_1$, how does one find the point $(x_1,y_1)$ so that…
sam wolfe
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How to know if a point is inside a circle?

Having a circle with the centre $(x_c, y_c)$ with the radius $r$ how to know whether a point $(x_p, y_p)$ is inside the circle?
Ivan
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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 required? Using the Wolfram Alpha site, this input…
Vincent Tan
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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?
andikat dennis
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False proof: $\pi = 4$, but why?

Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I know from Calculus, but I emphasize that I have…
ccbreen
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Dog bone-shaped curve: $|x|^x=|y|^y$

EDITED: Some of the questions are ansered, some aren't. EDITED: In order not to make this post too long, I posted another post which consists of more questions. Let $f$ be (almost) the implicit curve$$|x|^x=|y|^y$$ See the graph of the…
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Lampshade Geometry Problem

Today, I encountered a rather interesting problem in a waiting room: $\qquad \qquad \qquad \qquad$ Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my response below, I will…
Kaj Hansen
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What is the 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 $$x_1=r \cos \theta$$ $$x_2=r \sin \theta$$ For…
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Why are we justified in using the real numbers to do geometry?

Context: I'm taking a course in geometry (we see affine, projective, inversive, etc, geometries) in which our basic structure is a vector space, usually $\mathbb{R}^2$. It is very convenient, and also very useful, since I can then use geometry…
Olivier
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What is this curve formed by latticing lines from the $x$ and $y$ axes?

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 $n\rightarrow \infty$? Update: According to the answers this curve is not a part…
user108850
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How to calculate the intersection of two planes?

How to calculate the intersection of two planes ? These are the planes and the result is gonna be a line in $\Bbb R^3$: $x + 2y + z - 1 = 0$ $2x + 3y - 2z + 2 = 0$
user1111261
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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 $x$ coordinate of the vertex. I asked her why, and…
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If $AM$, $BN$, $CP$ are parallel chords in the circumcircle of equilateral $\triangle ABC$, then $\triangle MNP$ is also equilateral

Here's an interesting problem, and result, that I wish to share with the math community here at Math SE. The above problem has two methods. Pure geometry. A bit of angle chasing and standard results from circles help us arrive at the desired…
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