From One Collinearity to Another

The following theorem is due to Dao Thanh Oai (this is theorem 5 from his online paper):

Let collinear points $A_1,$ $B_1,$ $C_1$ lie on the sidelines $BC,$ $AC,$ and $AB$ of $\Delta ABC.$ Assume points $A_2,$ $B_2,$ $C_2$ are collinear with $A_1,$ $B_1,$ $C_1$ and define $A_3=BC\cap AA_2,$ $B_3=AC\cap BB_2,$ $C_3=AB\cap CC_2.$

From One Collinearity to Another

Assume also that


Then points $A_3,$ $B_3,$ $C_3$ are collinear.

Note that the required condition is satisfied when $A_2,$ $B_2,$ $C_2$ are the midpoints of $B_1C_1,$ $A_1C_1,$ and $A_1B_1,$ respectively.


Let's agree that below all the segments are directed.

In $\Delta B_1A_1C$ with a transversal $A_2AA_3,$ the Menelaus theorem gives


In $\Delta C_1BA_1$ with a transversal $AA_3A_2,$ we have


The latter two combine to prove


Similarly we obtain




Multiplying the three and taking into account the statement of Menelaus' theorem for $\Delta ABC$ and the transversal $C_1A_1B_1,$ i.e.,


we get after simplification,


The converse of Menelaus' theorem shows that points $A_3,$ $B_3,$ $C_3$ are indeed collinear.

Menelaus and Ceva