In a previous note the present author has given an account of the conjugations known as isogonal conjugation and isotomic conjugation with respect to a given triangle and has shown how to generalise the idea of conjugation involving a pair of points so that P(l, m, n) is q-conjugate, by means of a construction involving a transversal q, to the point (p2/l, q2/m, r2/n) and he has also demonstrated how to perform the geometrical constructions involved. When there is a self-conjugate point at the incentre I one has isogonal conjugation, and when there is a self-conjugate point at the centroid G one has isotomic conjugation.

Solid shapes which pack together to fill space cover a large and varied range. I recently found five such polyhedra which I have not seen described elsewhere. The names which I have used for them here are rather ferocious I am afraid, but that just happens to be the way the names of polyhedra work.

In 1355 the Italian professor of law Bartolus of Saxoferrato (1313-1357) wrote a treatise on the division of alluvial deposit. The problem he discussed is the following (figure 1). Some landowners, Bartolus calls them Gaius, Lucius, and Ticius, have neighbouring properties beside the bank of a river. The river deposits silt so that new land is formed at the riverside. How is this new fertile soil to be divided up ?

Morley's theorem states that the points of intersection of adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle, known as Morley's triangle. See Figure 1.

Concise proofs of the theorem are given in recent papers [1, 2]. A good picture of important previous work can be obtained by looking at [3, p. 1999], and examining references cited there.