Question 21
Points $A_1, A_2, \ldots, A_n$ are equally spaced on the side $BC$ of the triangle $ABC$ (so that $BA_1 = A_1A_2 = \ldots = A_{n-1}A_n = A_nC$). Similarly, points $B_1, B_2, \ldots, B_n$ are equally spaced on the side $CA$, and points $C_1, C_2, \ldots, C_n$ are equally spaced on the side $AB$. Show that
$$(AA_1^2 + AA_2^2 + \ldots + AA_n^2 + BB_1^2 + BB_2^2 + \ldots + BB_n^2 + CC_1^2 + \ldots + CC_n^2)$$
is a rational multiple of $$(AB^2 + BC^2 + CA^2).$$
$$(AA_1^2 + AA_2^2 + \ldots + AA_n^2 + BB_1^2 + BB_2^2 + \ldots + BB_n^2 + CC_1^2 + \ldots + CC_n^2)$$
is a rational multiple of $$(AB^2 + BC^2 + CA^2).$$