# 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.

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### 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?
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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?
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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 ...
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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?
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### Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
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### 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 ...
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### How do I prove this method of determining the sign for acute or obtuse angle bisector in the angle bisector formula works?

The formula for finding the angular bisectors of two lines $ax+by+c=0$ and $px+qy+r=0$ is: $$\frac{ax+by+c}{\sqrt{a^2+b^2}} = \pm\frac{px+qy+r}{\sqrt{p^2+q^2}}$$ I understand the proof of this formula ...
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### What Does Homogenisation Of An Equation Actually Mean?

For example, if we have a conic: $$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$$ What does homogenising this equation with another line (say $ax + by + c = 0$ ) actually mean? As in, what are the ...
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### 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 ...
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### 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 ...
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### 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 ...
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### What is the cone of the conic section?

Given the general (real valued) equation of a conic section: $$A x^2 + B xy + C y^2 + D x + E y + F = 0$$ Then what is the circular cone associated with it ? Is it unique ? And is there a way to ...
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