Example 2
The teacher asks the students to do a free search for a formula that relates lengh of sides and area of a triangle.
The student makes a sketch with GeoGebra:
Student introduces generic coordinates, gives names a, b, c to the side lenghts 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, in particular that
a^4 - 2a^2b^2 + b^4 - 2a^2c^2 - 2b^2c^2 + c^4 + 16S^2=0
> | GG:=Basis({a^2-((r-l)^2+s^2), b^2-(r^2+ s^2), c^2-l^2, l*s-2*S},plex(s,l,r,a,b,c,S));
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(3.2.1) |
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 that Heron's formula is equivalent to that one, naming p the semi-perimeter 2p-(a+b+c).
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> | GGG:=Basis({2*p-(a+b+c),
a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4 + 16*S^2 },tdeg(p,a,b,c,S)): NormalForm(S^2-p*(p-a)*(p-b)*(p-c), GGG, tdeg(p,a,b,c,S)); |
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(3.2.2) |
> | simplify(subs(p=(a+b+c)/2, 16*(S^2-(p*(p-a)*(p-b)*(p-c))))); |
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(3.2.3) |
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