Diagonal triangle of cyclic quadrangle and inversion in isotropic plane

In the paper the cyclic quadrangle and its diagonal triangle in the isotropic plane are studied. We prove that the nine-point conic of the quadrangle is a special hyperbola with the center in the centroid of the quadrangle. Furthermore, an inversion with respect to the circumcircle of the quadrangle and the absolute point as the pole is studied. We show that the nine-point conic of the quadrangle is mapped onto a 3-circular cubic, and that the Euler circle of the diagonal triangle is the inverse image of the circumcircle of the diagonal triangle. Another important result proved is that there is a unique invariant cubic passing through the vertices of the quadrangle. It is a 2-circular cubic and passes through the vertices of the diagonal triangle and its pedal triangle as well.