I want to address the question of how to determine whether or not four points are concyclic. In a previous post, this question reared its head and was solved by an introduction of the circumcentre of three of the points. In this post, I present a direct approach.

We prove the following:

Let be four points in . Then these four points are concyclic if and only if

For the proof, first suppose that the points all lie on the curve

The four equations we obtain give a linear dependence of the columns in the matrix under consideration, hence its determinant is zero.

Conversely, suppose that the determinant in question is zero. Then there is a nonzero vector in the nullspace of our matrix. If its first entry is nonzero, we can normalise it to be of the form and we obtain the circle that the four points lie on. If the first entry of this vector is equal to zero, then the four points must lie on a line, which after is all is just a degenerate circle, QED.

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