Question 131
1. For a convex hexagon $A B C D E F$ in which each pair of opposite sides is unequal, consider the following six statements:
$$
\begin{array}{ll}
\text{(a\_1) } A B \text{is parallel to this} D E ; & (a_2)\ A E = B D \\
(b_1)\ B C \text{is parallel to this} E F ; & (b_2)\ B F = C E \\
\text{(c\_1) } C D \text{is parallel to this} F A ; & (c_2)\ C A = D F
\end{array}
$$
(a) Show that if all the six statements are true, then the hexagon is cyclic (i.e., it can be inscribed in a circle).
(b) Prove that, in fact, any five of these six statements also imply that the hexagon is cyclic.
$$
\begin{array}{ll}
\text{(a\_1) } A B \text{is parallel to this} D E ; & (a_2)\ A E = B D \\
(b_1)\ B C \text{is parallel to this} E F ; & (b_2)\ B F = C E \\
\text{(c\_1) } C D \text{is parallel to this} F A ; & (c_2)\ C A = D F
\end{array}
$$
(a) Show that if all the six statements are true, then the hexagon is cyclic (i.e., it can be inscribed in a circle).
(b) Prove that, in fact, any five of these six statements also imply that the hexagon is cyclic.