Consider the circle circumscribed about the triangle ABC and note its center, 'O'.

It is easy to see immediately that a right-angled triangle ABC has been created.

Thus six plus the original equals the whole triangle ABC.

The three circles just constructed are also designated as epicycles of triangle ABC.

The generalisation begins with an arbitrary point P in the plane of a triangle ABC.

Applying sum of angles in triangle ABC it should be noted that .

Let X be any triangle center of the triangle ABC.

This line is the antiorthic axis of triangle ABC.

This line is the line at infinity in the plane of triangle ABC.

Invert with respect to the incircle of triangle ABC.