The foot of the perpendicular from (a,b,c) on the line $x=y=z$ is the point $(r,r,r)$ where

Dr s of line is 1,1,1

dr s of line joining (a,b,c)&(r,r,r) isr-a,r-b,r-c

both are perpendicular 

1(r-a)+1(r-b)+1(r-c)=0