Belarus — Selection and Training Session

Let the coordinates of the given points be $A(a; 1/a)$, $B(b; 1/b)$, $C(c; 1/c)$, $A_1(a_1; 1/a_1)$, $B_1(b_1; 1/b_1)$, $C_1(c_1; 1/c_1)$. It is easy to calculate the slope of the line $AB$: $k = -1/(ab)$. Similarly, the slopes of $A_1B_1$, $BC$, $B_1C_1$ are $-1/(a_1b_1)$, $-1/(bc)$, $-1/(b_1c_1)$, respectively. Now, the conditions $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$ are equivalent to the equalities $-1/(ab) = -1/(a_1b_1)$ and $-1/(bc) = -1/(b_1c_1)$. These equalities imply the equality $-1/(ac_1) = -1/(a_1c)$ which is equivalent to the condition $AC_1 \parallel A_1C$.