Geometry Explorer: Combining Dynamic
Geometry, Automated Geometry Theorem Proving
and Diagrammatic Proofs
Sean Wilson and Jacques D. Fleuriot
Sean Wilson and Jacques D. Fleuriot
ARW-05
Geometry Explorer
1
Motivational Example
(Nine Point Circle Theorem) Let AD be the altitude on BC and let the
midpoints of the sides AB, BC and CA of △ABC be E, F and G respectively.
Show that D, E, F and G are on the same circle.
Sean Wilson and Jacques D. Fleuriot
ARW-05
Geometry Explorer
2
Full-Angles
• The full-angle between the ordered pair of lines u and v is written as 6 [u, v]
and is the anti-clockwise rotation required to make u parallel to v. For
example, measuring the full-angle 6 [AB, CD]:
• If u k v then 6 [u, v] = 6 [0] is a constant.
• If u ⊥ v then 6 [u, v] = 6 [1] is a constant.
Sean Wilson and Jacques D. Fleuriot
ARW-05
Geometry Explorer
3
The Full-Angle Method
1. In predicate form, the hypotheses is put into a so-called Geometry Information
Basis (GIB).
2. Exhaustive forward-chaining is applied to the GIB to discover new facts, using
rules such as:
F3 If M and N are the midpoints of AB and AC respectively then M N k BC.
F5 If O is the midpoint of CA and AB ⊥ BC then O is the circumcenter of
△ABC.
Sean Wilson and Jacques D. Fleuriot
ARW-05
Geometry Explorer
4
The Full-Angle Method (Continued)
3. The theorem conjecture is represented as 6 [0] =
P
fi, where fi is a full-angle.
4. Full-angles are replaced with equal expressions using conditional rewrite rules
such as:
R1
R2
R6
6
[AB, CD] = 6 [AB, EF ] if CD k EF .
[AB, CD] = 6 [AB, EF ] + 6 [1] if CD ⊥ EF .
6
6
[AB, BC] = 6 [AD, CD] if A, B, C and D are cyclic.
5. A search algorithm is used to find a sequence of rewrites that transforms the
full-angle equation to 6 [0] = 6 [0], giving a backward-chaining proof.
Sean Wilson and Jacques D. Fleuriot
ARW-05
Geometry Explorer
5
A Diagrammatic Forward-Chaining Proof
Sean Wilson and Jacques D. Fleuriot
ARW-05
Geometry Explorer
6
A Diagrammatic Backward-Chaining Proof
Sean Wilson and Jacques D. Fleuriot
ARW-05
Geometry Explorer