## Problem

Given 4 points in plane, how to determine they fall on a common circle?

There are lots of methods can do this, and here I introduce the determinant method, which is very straightforward, elegant and can easily generalized to quadratic curve.

**The following content is inspired by this.**

First of all, we can write the circle equation defined by by a determinant:

**Proposition 1:**

When , is inside the circle defined by ，and outside when , right on the circle when ,which means a circle go through simultaneously.

Proof:

Notice that:

( is the minor of the matrix)

and if we regard as a moving point, then is actually an equation of a circle:i.e. the circle locate at

with radius .When the circle equation become

So it’s clear that if fall inside that circle, then and vice versa.

**Note**: According to Cramer’s rule, is exactly the solution of the equation

And this 3 parameters perfectly define a circle pass through .One can easily verify when is or , by the equation above.

## Generalization

We can generalize this method to the question of **determine whether 6 points lies on some quadratic curve **

Any equation of quadratic curve can write as (and again, we regard as a moving point )

and expand the last row of :

Definition.

A point isinsideof a quadratic curve if there are no tangent of that curve pass through . Andoutsideif there is at least 1 tangent pass through it.

And still we can tell whether lies in or out of by check the sign of .

Let be the coefficient of the quadratic equation, we have:

Proposition

The point lies inside of a quadratic curve If has the same sign with the determinant

But unfortunately I can’t give the proof **temporarily**. If you have any idea, please let me know, thanks in ahead!

### Check the shape of the curve

If we want to check whether the six point lies on eclipse or hyperbola etc. we can simply check the sign of **geometry invariants**

Symbols.

Theorem.

The curve is classified by the following criterion:

Eclipse :

Parabola:

Hyperbola:

Two parallel lines :

Two intersecting lines:

A line( two line coincide):

Imaginary ellipse :

Point(two im. line intersect at real plane) :

Two parallel imaginary lines: