Bringing More Intelligence to Dynamic
Geometry by Using Symbolic Computation
Francisco BotanaUniv. Vigo (Spain)
http://webs.uvigo.es/fbotana
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Outline
DG problems: continuity, loci, proofGDI (Intelligent Dynamic Geometry)(partial) Solutions to DG problemswebDiscovery: breaking up the Algebra and Geometry partsIntercommunication
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Discontinuity ...
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... or non determinism
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Geometric loci(sampling approach pitfalls)
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Geometric loci(a posteriori conditions)
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Proof
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GDIGeometría Dinámica Inteligente
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GDI(textfile)
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GDI(geometric properties)
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GDI(CoCoA)
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GDI(discovering and/or proving)
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GDI(discovering)
The triangle ABX:
1.- Is impossible
2.- Is isosceles
3.- Is equilateral
4.- Is a right triangle
5.- None of the above
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GDI(discovering)
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GDI(discovering)
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webDiscovery
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webDiscovery(An Euler’s formula)
Midpoint(X1,A,B)
Midpoint(X2,A,C)
Aligned(X3,A,B)
Aligned(X4,A,C)
Aligned(X5,B,C)
Perp(A,B,X1,Ci)
Perp(A,C,X2,Ci)
Perp(A,B,I,X3)
Perp(A,C,I,X4)
Perp(B,C,I,X5)
d(I,X3)=d(I,X4)
d(I,X3)=d(I,X5)
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webDiscovery(An Euler’s formula)
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webDiscovery(An Euler’s formula)
Use R::=Q[hu[5..10]drc];
Elim(h..u[10],Ideal(h u[6]-1,
d^2-((u[9]-u[7])^2+(u[10]-u[8])^2),
r-u[8],
c^2-(u[9]^2+u[10]^2),
2u[9] - 1,
...
));
Ideal(1/2d^4 - d^2c^2 - 2r^2c^2 + 1/2c^4)
-------------------------------
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Intercommunication
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Intercommunication
{1} Point(-31,227)[hidden];
{2} Point(590,227)[hidden];
{3} Point(206,107);
{4} Point(296,107);
{5} Line(2,1)[black];
{6} Segment(4,3)[black];
{7} Point on object(5,0.32528180)[label('A')];
{8} Point on object(5,0.58454108)[label('B')];
{9} Circle by radius(7,6)[black];
{10} Circle by radius(8,6)[black];
{11} Intersect2(10,9)[traced,label('P')];
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References
http://webs.uvigo.es/fbotana
Thank you.