The Sherman line of a triangle is the additional line, distinct from its three sidelines, that is tangent to the incircle and cuts the circumcircle in a chord whose midpoint lies on the nine-point circle. The chord is called the fourth side of a triangle (Sherman 1993; Yiu 2012, p. 219).
In the illustration, the Sherman line extends beyond the endpoints of the chord 
. Here 
is the midpoint of the chord,
and 
is its contact point with the incircle.
The radius 
from the incenter 
is perpendicular to 
.
Let 
, 
, and 
be the side lengths of the reference
triangle, and put
![f(a,b,c)=(b-c)[a^2(b+c)-2abc-(b+c)(b-c)^2].](/images/equations/ShermanLine/NumberedEquation1.svg) |
When 
,
the Sherman line has equation
 |
Here 
are barycentric coordinates with respect
to the reference triangle (Yiu 2012, pp. 222-223).
For a triangle without a right angle, the Sherman line is also the fourth common tangent line of the incircle and the Macbeath inconic (Yiu 2012, pp. 221-222). The nonzero condition excludes cases in which the fourth line coincides with a sideline, which can occur even for a scalene triangle. For an equilateral triangle, the incircle and nine-point circle coincide, and infinitely many tangent lines have the required chord property, so there is no distinguished fourth line.
