Bringing More Intelligence to Dynamic

Geometry by Using Symbolic Computation

Francisco BotanaUniv. Vigo (Spain)

http://webs.uvigo.es/fbotana

04/18/23 2

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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Intercommunication

nash.sip.ucm.es/CabriOM

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References

http://webs.uvigo.es/fbotana

Thank you.