The Fourth Side of a Triangle

Abstract

One of the earliest geometrical facts we learn is that a triangle has three sides; proposing a fourth side is about on the same level as suggesting that two and two are not four. Nevertheless, the fourth 'side' found here, whilst not joining any of the vertices of the triangle, does arise in a natural way from the properties of the triangle. For this to be seen, the sides must be considered from the perspective of certain circles associated with the triangle-the circumcircle, the incircle, and the nine-point circle. The nine-point circle is the circle through the three midpoints of the sides; it also passes through the feet of the altitudes, and the mid-points of the lines joining the orthocentre to the vertices. It is also known as the Euler circle or the Feuerbach circle. From the point of view of the three circles, the sides satisfy the following conditions: a) Each side has its endpoints on the circumcircle; b) each side has its midpoint on the nine-point circle; and c) each side touches the incircle. In looking for line segments that satisfy these conditions, I was amazed to find that, for most triangles, there were not three such segments but four. Three of them were clearly the three sides as we know them; what else was there to call the last but the fourth side of the triangle?