**Three simple instances**

__Example 1____
__
The angle defined by a diameter of a circle from a point of the same circle is always a right angle.

*circle of center (l/2,0), passing thru (0,0)

*(s,r) on circle: (s-l/2)^2+r^2-(l/2)^2=0

*(s,r) ^ (s-l,r), ie. s(s-l)+r^2=0

In fact,

s(s-l)+r^2 = (s-l/2)^2+r^2-(l/2)^2

write(ring)

Ring( "ring name:" R ;

"characteristic:" 0 ;

"variables:" slr ;

"weights:" 1 , 1 , 1 ;

"ordering:" DEGREVLEX );

NormalForm(s(s-l)+r^2,Ideal((s-l/2)^2+r^2-(l/2)^2) )

0

The teacher asks the students to search at their own for a formula that relates lengh of sides and area of a triangle.

¥The student makes a sketch with MacDraw

¥Student gives names A, B, C to the side lenghts.

¥Computer assigns coordinates to the vertices

and introduces in the symbolic program the equations for A, B, C.

A^2-((r-l)^2+s^2)=0

B^2-(r^2+ s^2)=0

C^2-l^2=0

¥Computer asks the student: what are you searching for?

¥Student answers: formula relating area S of a triangle and A, B, C.

¥Computer asks: what is an area?

¥Student introduces the definition S=ls/2 (or something like)

¥Computer passes the new information to the symbolic program and gets all possible conclusions involving just A, B, C, S,

Use R::=Q[r,s,l,A,B,C,S];

r..l

[r, s, l]

Elim(r..l, Ideal(A^2-((r-l)^2+s^2), B^2-(r^2+ s^2), C^2-l^2, ls-2S))

Ideal( A^4 - 2A^2B^2 + B^4 - 2A^2C^2 - 2B^2C^2 + C^4 + 16S^2 );

¥So the students tell the teacher that:

"sixteen times the square of the area is equal to two times the square of A times the square of B, plus two times the square of A times the square of C, plus two times the square of B times the square of C, minus the sum of the fourth powers of A,B,C."

¥The teacher is perplexed. Turns on the pocket calculator and verifies whether Heron's formula is equivalent to that one, naming ÒpÓ the semi-perimeter: 2p-(A+B+C).

NormalForm(S^2-p(p-A)(p-B)(p-C),

Ideal(2p-(A+B+C),

A^4 - 2A^2B^2 + B^4 - 2A^2C^2 - 2B^2C^2 + C^4 + 16S^2 ))

0

For any given triangle, we claim the circumcenter lies on one side.

(x,y) circumcenter, ie.

x^2+y^2-(x-e)^2-y^2,

x^2+y^2-(x-a)^2-(y-b)^2

Now we claim it lies on the horizontal side, say,

T

¥ Theorem
**H==>T**
is false in general,

since the negation of H==>T,
**
H &ÂT**
is non-empty:

NormalForm(1,Ideal(x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, yt-1))

1;

¥ So we look for special cases where the theorem could hold:

Elim(t..y,Ideal(x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, y))

Ideal( a^2e + b^2e - ae^2 );

¥We learn that if there are chances for the theorem to hold, it should happen that

a^2e + b^2e - ae^2=0

for instance, we could start with having a right triangle, ie. with (a,b) perpendicular to (a-e, b).

¥Then we continue trying with this set of enlarged hypotheses....