Point defined as a triangle center
Diagram of the Schiffler Point Triangle △ABC
Lines joining the midpoints of each angle bisector to the vertices of △ABC
Lines perpendicular to each angle bisector at their midpoints
In geometry , the Schiffler point of a triangle is a triangle center , a point defined from the triangle that is equivariant under Euclidean transformations of the triangle. This point was first defined and investigated by Schiffler et al. (1985).
Definition
A triangle △ABC with the incenter I has its Schiffler point at the point of concurrence of the Euler lines of the four triangles △BCI , △CAI , △ABI , △ABC . Schiffler's theorem states that these four lines all meet at a single point.
Coordinates
Trilinear coordinates for the Schiffler point are
1
cos
B
+
cos
C
:
1
cos
C
+
cos
A
:
1
cos
A
+
cos
B
{\displaystyle {\frac {1}{\cos B+\cos C)):{\frac {1}{\cos C+\cos A)):{\frac {1}{\cos A+\cos B))}
or, equivalently,
b
+
c
−
a
b
+
c
:
c
+
a
−
b
c
+
a
:
a
+
b
−
c
a
+
b
{\displaystyle {\frac {b+c-a}{b+c)):{\frac {c+a-b}{c+a)):{\frac {a+b-c}{a+b))}
where a, b, c denote the side lengths of triangle △ABC .
