# Menelaus’s Theorem of Collinearity

Menelaus’s Theorem: If three points (L, M, and N) are collinear (meaning that they lie on the same line), then the following relationship holds true: ${\color{red} \frac{AL}{LC}\cdot\frac{BM}{MA}\cdot\frac{CN}{NB}=1}$.

## Proof

To prove Menelaus’s Theorem, we draw perpendicular lines from the vertices of ΔABC to the line LN. From the similar triangles ΔALU and ΔCLW, we have the following relationship:

(i) ${\color{red}\frac{AL}{-CL}=\frac{AL}{LC}=-\frac{AU}{CW}}$

From the similar triangles ΔWCN and ΔVBN, we have the following relationship:

(ii) ${\color{red}\frac{CN}{-BN}=\frac{CN}{NB}=-\frac{CW}{BV} }$

From similar triangles ΔVBM and ΔUAN, we have the following relationship:

(iii) ${\color{red}\frac{BM}{-AM}=\frac{BM}{MA}=-\frac{BV}{AU} }$

Multiplying (i), (ii), and (iii), we obtain:

(iv) ${\color{red} \frac{AL}{LC}\cdot\frac{BM}{MA}\cdot\frac{CN}{NB}=}$ ${\color{red} -\frac{AU}{CW}\cdot-\frac{CW}{BV}\cdot-\frac{BV}{AU}= -1 }$

The negative sign is necessary when considering the order of writing the segment. For example, segment LC = –CL.

This completes the proof. The converse of this theorem is also true. If the relations (iv) is true, then point L, M, and N must be collinear.