The half-altitude triangle 
of a reference
triangle 
is defined by letting 
be the midpoint between vertex
and the foot of the 
-altitude on side 
, and similarly defining 
and 
(Kimberling 1998, pp. 172-173).
The half-altitude triangle has trilinear vertex matrix
![[1 cosC cosB; cosC 1 cosA; cosB cosA 1].](/images/equations/Half-AltitudeTriangle/NumberedEquation1.svg) |
It has area
 |
where 
is the area of 
.
The circumcircle of the half-altitude triangle is the half-altitude circle.
