The fourth side of a triangle is the additional chord of its circumcircle, distinct from its three sides, that is tangent to its incircle and has its midpoint on the nine-point circle (Sherman 1993; Yiu 2012, p. 219). Its infinite supporting line is the Sherman line.
In the illustration, the fourth side is the finite line segment 
,
with midpoint 
and incircle contact point 
. The radius 
from the incenter 
is perpendicular
to 
.
When the Sherman line coincides with a sideline, it does not determine a distinct fourth side. For an equilateral triangle, infinitely many chords have the required properties, so there is no distinguished fourth side.
