000RM.dvi

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

)