Given a triangle 
and the excentral triangle 
, define the 
-vertex of the hexyl triangle as the point in which the perpendicular
to 
through the excenter 
meets the perpendicular to 
through the excenter 
, and similarly define 
and 
. Then 
is known as the hexyl triangle of 
, and 
forms a hexagon
with parallel sides (Kimberling 1998 pp. 79 and 172).
The hexyl triangle has trilinear vertex matrix
![[x+y+z+1 x+y-z-1 x-y+z-1; x+y-z-1 x+y+z+1 -x+y+z-1; x-y+z-1 -x+y+z-1 x+y+z+1],](/images/equations/HexylTriangle/NumberedEquation1.svg) |
(1)
|
where 
,
, and 
(Kimberling 1998, p. 172).
It has side lengths
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
and area
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
where 
is the area of the reference triangle, 
is the circumradius,
and 
is the inradius. It therefore has the same side lengths
and area as the excentral triangle.
The Cevians triangles with Cevian points corresponding to Kimberling centers 
with 
, 20, 21, 27, 63, and 84 are perspective to the hexyl triangle.
That anticevian triangles and antipedal triangles corresponding to Kimberling centers
for 
,
9, 19, 40, 57, 63, 84, 610, 1712, and 2184 are also perspective to the hexyl triangle
In fact, any point on the trilinear cubic
![sum_(cyclic)[betagamma(S_Cbeta-S_Bgamma)]=0](/images/equations/HexylTriangle/NumberedEquation2.svg) |
(8)
|
has an anticevian and antipedal triangle that are perspective with the hexyl triangle (P. Moses, pers. comm., Feb. 3, 2005).
The circumcircle of the hexyl triangle is the hexyl circle.
The triangle centroid of the hexyl triangle is the point with triangle center function
 |
(9)
|
which is not a Kimberling center.
The following table gives the centers of the hexyl triangle in terms of the centers of the reference triangle for Kimberling centers
with 
.
 | center of hexyl triangle |  | center of reference triangle |
 | circumcenter |  | incenter |
 | orthocenter |  | Bevan point |
 | nine-point center |  | circumcenter |
 | triangle centroid of orthic triangle |  | triangle
centroid of the antipedal triangle of  |
 | orthocenter of orthic triangle |  | de Longchamps point |
 | symmedian point of orthic triangle |  | reflection
of 
in  |
 | Prasolov point |  | isogonal
conjugate of  |
 | Jerabek antipode |  | anticomplement of Feuerbach point |
 | Kiepert antipode |  | psi(incenter, symmedian point) |
 | Kiepert center |  | antipode
of  |
 |  -of-medial triangle |  | lambda(incenter, triangle centroid) |
 | Jerabek center |  | antipode
of  |
