A Note on the Need for Radical Membership Checking in Mechanical Theorem Proving in Geometry

In his famous 1988 book "Mechanical Geometry Theorem Proving", Shang-Ching Chou details a method with two steps to check whether a geometric theorem is "generally true" or not using Groebner bases. The second step consists of checking the membership of the thesis polynomial to the radical (J) of a certain ideal (L). Chou mentions: "However, for all theorems we have found in practice, J=L." In his 2007 book "Selected topics in geometry with classical vs. computer proving", Pavel Pech shows a beautiful example where checking the radical membership, not only the ideal membership, is required. Using a kind of "reverse engineering" we shall show how to easily find examples of theorems where to check the radical membership is required. The idea is just to construct a thesis involving an ideal such that the ideal of its variety is not itself (therefore, the ideal is not equal to its radical).