Suppose the lengths of the three sides of triangle $ABC$ are 9, 12 and 15 respectively. Divide each side into $n$ ($>=2$) 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 triangle $ABC$ to the $n-1$ division points lying on its opposite side. If $S$ is an integer, find all possible positive integer $n$, with detailed answers.
My solution:
I divided each side into three parts, with two division points each. The measurement of each division line is : a,b,c,d,e,f.
( $a$ and $b$ are distances to line of size 15,
$c$ and $d$ are distances to line of size 12,
$e$ and $f$ are distances to line of size 9)
Then I get the following equations:
$a^2$+$b^2$=$S$
$c^2$+$d^2$=$S$
$e^2$+$f^2$=$S$
Is there a smarter way to solve this problem?
Please advise.