Discovering and proving geometric inequalities by CAS (pdf)

We also used this theory to carry out constructions of geometric objects which have given
properties and which are not easy to construct by using ruler and compass.
Automatic theorem proving
Automatic theorem proving is concerned with geometry statements of equality type, which
are of the kind H ⇒ C, whereH is the set of hypotheses and C the conclusion.
First we express the geometric problem in an algebraic form. Let K be a field of characteristic
0, e.g. the field of rational numbers, and L be an algebraically closed which contains
K, e.g. the field of complex numbers. The first stage of automatic theorem proving is
characterized by establishing the set of hypotheses H whose algebraic form are polynomial
equations
h1(x1,x2,...,xn) =0, h2(x1,x2,...,xn) =0,...,hr(x1,x2,...,xn) =0
and the conclusion C, which is expressed by the polynomial equation
c(x1,x2,...,xn) =0,
where h1,h2,...,hr,c∈ K[x1,x2,...,xn]. Thus the algebraic form of the statement is
∀x ∈ L n , h1(x) =0, h2(x) =0, ..., hr(x) =0 ⇒ c(x) =0, (1)
where we write x instead x1,x2,...,xn. The objective of the next step is a verification
of (1). We are to decide whether the conclusion follows from the hypotheses or, which is
the same, to decide whether the zero set of the conclusion C contains the zero set of the
hypotheses H, i.e., Zero (H) ⊂ Zero (C).
By the well-known Hilbert Nullstellensatz Theorem, the statement (1) is true iff 1 belongs
Figure 1: H ⇒ C ⇔ Zero(H) ⊂ Zero(C)
to the ideal (h1,h2,...,hr,ct − 1) of the hypotheses polynomials and negated conclusion.
However for most geometry problems it suffices to show that c belongs to the ideal
(h1,h2,...,hr). If yes we say that a statement (1) is generally true.
See a nice book [3] for further study.
Parallelogram law
To demonstrate the base of automatic theorem proving we start with investigation of
equalities between diagonals holding for various types of quadrilaterals. We will explore
the equality between the sum of squares of sides and the sum of squares of diagonals of a
parallelogram. This equality is known as the parallelogram law [6]: