For a real number p, let A(p,2p+3) and B(5p−2,−p+7) be two points. A point D has coordinates (513p−4,54p+23).
(i) Prove that, for every real value of p, the point D divides the segment AB internally in the constant ratio 2:3.
(ii) Taking p=4, write the numerical coordinates of A, B and D.
(iii) For p=4, compute the lengths AD and DB and verify that their ratio equals 2:3.