The hypocycloid of STEINER, we have seen above, is an algebraic curve of order four.

In the special case of an equilateral triangle ABC the two conics c' and c'' coincide with the circumcircle u of the triangle and the WALLACE-line [PQR] and the lines [P'Q'R'] and [P''Q''R''] with respect to the point X form an equilateral triangle STU with the following properties, see figure 8:

• The vertex S lies on the circumcircle u of the triangle ABC.
• The other two vertices T and U describe the same closed symmetrical rational curve of order four with three axes and the three nodes A, B, C.
• On the other hand the point X always lies on the circumcircle v of the triangle STU and the circumcenter V of triangle STU describes a closed rational curve of order four as well.

These curves we create with CINDERELLA very quickly using the locus mode with the mover point X. To determine the order of the curves we have to make algebraic considerations by ourselves.
The incidence of S with the circumcircle u of the triangle ABC and the incidence of X with the circumcircle v of the triangle STU are automatically reported by CINDERELLA in the information console .