Proof.

First, consider the following diagram.

From the diagram above, it is evident that \(X,Y,Z\) are collinear if and only if Menelaus’ Theorem holds for \(\triangle {O_1O_2O_3}\). In other words, it suffices to show that the following equation holds. \[ \frac {O_2 X}{O_1 X} \cdot \frac {O_3 Y}{O_2 Y} \cdot \frac {O_1 Z}{O_3 Z} = 1 \] Note that \(\frac {O_2 X}{O_1 X} = \frac {R_2}{R_1}\) by similar triangle. Similarly, \(\frac {O_3 Y}{O_2 Y} = \frac {R_3}{R_2}\) and \(\frac {O_1 Z}{O_3 Z} = \frac {R_1}{R_3}\). Because \(\frac {R_2}{R_1} \cdot \frac {R_3}{R_2} \cdot \frac {R_1}{R_3} = 1\), the theorem holds.