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 $A_1{B}_1$, $BC$, $B_1{C}_1$ are $-1/(a_1{b}_1)$, $-1/(bc)$, $-1/(b_1{c}_1)$, respectively. Now, the conditions $AB\parallel A_1{B}_1$ and $BC\parallel B_1{C}_1$ are equivalent to the equalities $-1/(ab)=-1/(a_1{b}_1)$ and $-1/(bc)=-1/(b_1{c}_1)$. These equalities imply the equality $-1/(a{c}_1)=-1/(a_{1}c)$ which is equivalent to the condition $A{C}_1\parallel A_{1}C$.