Nearly a month of the summer term remains to be got through by the somewhat jaded boys and girls after the Higher School examination is over. For many years I have been casting round for stimulating topics to discuss with my more or less advanced mathematicians, and this paper is concerned with my latest find. Its title refers to the four tangents at the points of contact of the nine-points circle of a triangle with the four circles touching the sides. I am aware that a great many properties of the points of contact themselves have been investigated, and I cannot flatter myself that I have found out anything new about the figure formed by the tangents thereat. But the literature of any branch of mathematics is so vast that it seemed best for me, instead of hunting in books, to try to find out what I could in my own way. The effort to do so resulted in a delightful mathematical adventure, which has enabled me to devise something to bring out to our pupils what analytical geometry really is about.