In most of our episodes I am showing some results which are ‘well known’ to mathematicians but may come as an interesting surprise to others. We shall be looking at four collinear centres of a triangle that divide their common line in a pleasing ratio. The ‘common line’ is the Euler Line marked in red in the nearby diagram. This is a classical result (18/19C). I had a surprise myself while preparing this episode to discover there had been a substantial renewal of interest in these (and very many other) ‘centres’ of a triangle in the 1980s – find out more in Tuesday evening’s episode!