how to find the centroid, circumcenter and orthocenter of a triangle?

If ( m , n ) ; ( u , v ) and ( p , q ) are the vertices of a triangle then
( (m+u+p) /3 , (n+v+q) /3 ) is the centroid ; if ( k , s ) is the circumcenter
then k and s can be found by solving ,
(n - v) / ( u - m ) = ( 2k - m - u ) / ( 2s - v - n ) ..... ( i )
( v - q ) / ( p - u ) = ( 2k - p - u ) / ( 2s - v - q ) ..... ( ii )
if ( w , f ) is the orthocenter then w and f can be found by solving ,
( f - n ) / ( w - m ) = ( u - p ) / ( q - v ) ..... ( 1 )
( f - v ) / ( w - u ) = ( m - p ) / ( q - n ) ..... ( 2 )