SUPPOSE THE LENGTHS OF THE THREE SIDES OF ABC∆ ARE 9, 12 AND 15...

2.

Suppose the lengths of the three sides of ABC

∆

are 9, 12 and 15 respectively. Divide

each side into ( 2)

n

≥

segments of equal length, with

n

−

1

division points, and let S

be the sum of the square of the distances from each of 3 vertices of ABC

∆

to the

1

n

−

division points lying on its opposite side.

If S is an integer, find all possible positive integer n, with detailed answers.