How to determine whether 4 points are concyclic

Standard

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 is inside of a quadratic curve if there are no tangent of that curve pass through . And outside if 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: