Let #L_1# be the perpendicular bisector of the line #AB#,
#=> L_1# passes through #AB#'s midpoint at #90^@#,
let #M# be the midpoint of #AB#,
#=> M=((4+10)/2,(6+2)/2)=(7,4)#
slope of #AB=m_(AB)=(2-6)/(10-4)=-2/3#
recall that if two non-vertical lines are perpendicualr, the product of their slopes is #-1#
#=># slope of #L_1=3/2#
#=># equation of #L_1#:
#y-4=3/2(x-7)#
#=> color(red)(2y=3x-13 ------ Eq(1))#
Let #L_2# be the line parallel to #AB# through #D(3,11)#
slope of #L_2=m_(L_2)=m_(AB)=-2/3#
equation of #L_2# :
#y-11=-2/3(x-3)#
#=> color(red)(3y=-2x+39 ------ Eq(2))#
#Eq(1)xx3 => color(red)(6y=9x-39 ------ Eq(3))#
#Eq(2)xx2 => color(red)(6y=-4x+78 ------ Eq(4)#
#Eq(3)-Eq(4) => 0=13x-117#,
# => x=9#
#=> y=7#
#=># coordinates of #C=(9,7)#
