706 Routh and Ceva theorems
Example: Gergonne point
IfX,Y,Zare the points of tangency of the incircle with the sidelines,
the linesAX,BY,CZintersect at the Gergonne point.
s−b s−c
s−c
s−a
s−a
s−b
A
B C
P
Z
X
Y
X =(0:s−c:s−b)=
(
0:s−^1 b:s−^1 c
)
Y =(s−c:0:s−a)=
( 1
s−a:0:
1
s−c
)
Z =(s−b:s−a:0) =
( 1
s−a:
1
s−b:0
)
P ==
( 1
s−a:
1
s−b:
1
s−c