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.


[Maple Metafile]
Hypothesis:
*circle of center (l/2,0), passing thru (0,0)
*(s,r) on circle: (s-l/2)^2+r^2-(l/2)^2=0

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

Proof
Is s(s-l)+r^2 in Ideal((s-l/2)^2+r^2-(l/2)^2)) ?

In fact,
s(s-l)+r^2 = (s-l/2)^2+r^2-(l/2)^2


CoCoA session
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



Example 2
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

[Maple Metafile]


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

•Computer assigns coordinates to the vertices

[Maple Metafile]



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,

CoCoA session

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 “p” 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



Example 3
For any given triangle, we claim the circumcenter lies on one side.

[Maple Metafile]

(x,y) circumcenter, ie.

H :
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
: y=0


• Theorem
H==>T is false in general,
since the negation of H==>T,

H &¬T
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....