In the configuration, illustrated in Figure 1, ABC is a triangle with I1; I2, I3 the excentres opposite A, B, C respectively. The triangles I1BC, I2CA, I3AB are denoted by T1, T2, T3 respectively. Ok, Hk, are the circumcentre, orthocentre respectively of triangle Tk, k = 1, 2, 3. Lk is the Euler line of Tk, k = 1, 2, 3. Define the point Ek (t) on Lk by the equation

For several years, the notion of ‘mathematical proof’ has been at the centre of much discussion, writing and educational research. To a very large extent the discussions have focused on the nature of the acceptability of the reasons for a mathematical conclusion. Admittedly there have been contributions concerning the role of proof in mathematics and also the mechanics of pupils‘ proof behaviour, including works which discuss both affective and pedagogical aspects, but the dominant context has usually been one of validation. That is to say, discussions have largely been in the domain of what ‘counts as a proof’ and for whom. However, I claim that there is another dimension to proof that is of central importance in the intellectual development of students at school and university. This relates to the phenomenon in mathematical thinking in which a section of mathematical structure is mentally compressed into a single unit, small enough to fit into the conscious focus of attention at a given time, and possessing an interiority which is able to both guide manipulation of the unit and also be subsequently expanded without loss of detail. This article discusses the role of this phenomenon in the manipulation of mathematical statements involved in proof. An analysis in these terms of the dynamics of certain types of mathematical proof suggests both an explanation for students’ difficulties and a possible means of help in some cases.