We use complex vectors in geometrical reasoning, specifically automated theorem proving. The calculations are embedded in
Clifford algebras, but commutative polynomial techniques can be used. Using the Gröbner basis package in the computer algebra
package Maple, this approach is shown to have efficiency benefits in situations where distance and angle relations amongst geometrical
entities predominate.
Acknowledgements We thank the anonymous referees of the first draft of this paper for their helpful suggestions. We would also like to thank
Desmond Fearnley-Sander for introducing us to the concept of automated theorem proving in geometry and many discussions over
the years on that topic.