Question

what are the coordinates of the three points that divide the line segment joining P(a,b) and Q(c,d) into four equal parts ...textbook answer is (3a+c/4,3b+d/4) is one of the three coordinats

Answer 1

Use the point of division formula.

Let the divided segment be
PABCQ

Then
r1=PA/PQ=1/4
r2=PB/PQ=2/4
r3=PC/PQ=3/4

Applying the point of division formula to points A, B and C:
A(a+r1(c-a), b+r2(d-b))
B(a+r2(c-a), b+r2(d-b))
C(a+r3(c-a), b+r3(d-b))

For more explanations and examples, see:
http://www.mathwords.com/p/point_of_division_formula.htm